Deriving all of modern physics from a single meta-evolutionary process
By Vladimir Ilinov
Introduction
Everything in reality emerges from a single meta-evolutionary process: Coherence. Assume this process and you can derive and unify spacetime (GR), quantum mechanics, and the Standard Model from first principles.
The Law of Coherence
A^(***)in arg max_(A)ubrace(i n f_(p inP)Hold(Pi _(A),K_(A,p)Pi _(A))ubrace)_("keeps working when poked")-lambda_(fuel)B_(fuel)(A)-lambda_(cpx)B_(cpx)(A)-lambda_(leak)B_(leak)(A)A^{\star} \in \arg\max_{A}\; \underbrace{\inf_{p\in\mathcal P}\; Hold\!\big(\Pi_A,\;K_{A,p}\,\Pi_A\big)}_{\text{keeps working when poked}}
\;-
\;\lambda_{\rm fuel}\,B_{\rm fuel}(A)
\;-
\;\lambda_{\rm cpx}\,B_{\rm cpx}(A)
\;-
\;\lambda_{\rm leak}\,B_{\rm leak}(A)
Pick the pattern that still works after every shove, charging it for fuel, complexity, and leakage. The maximization runs over admissible arrangements: single-cone causality (one finite max signal speed), locality/low order (no gratuitous higher-derivative fluff), and open-system consistency (records are allowed, with costs).
Pokes
Interfaces with neighbors. Pokes alter observables; only pre vs post matters: compare Pi _(A)\Pi_A to K_(A,p)Pi _(A)K_{A,p} \Pi_A. Content-agnostic; effect is what’s measured.
Budgets
Budget
Explanation
Throughput (fuel / energy)
Nothing runs for free.
Complexity (moving parts / derivatives / dependencies)
Higher-order baggage is taxed; the Gamma\Gamma-limit favors the simplest stable law.
Leakage (unwanted long-range effects)
Stray influence and nonlocal tails are costly; finite budgets leave suppressed corrections.
Patterns that survive pokes at minimal total cost persist. Others are used as fuel.
Coherence derives all known physics
Concept
Explanation
Lorentzian spacetime (the arena)
One signal cone =>\Rightarrow a two-sheeted light-cone =>\Rightarrow(1,3)(1,3) Lorentzian metric. Geometry is selected, not assumed.
Einstein’s law (the backbone)
Penalize higher derivatives; Gamma\Gamma-convergence yields second-order dynamics. In 4D, the unique survivor is Einstein–Hilbert; higher-curvature terms persist only as algebraically suppressed editsa_(i)(epsi)=O(epsi^(eta))a_i(\varepsilon)=\mathcal{O}(\varepsilon^{\eta}).
Quantum update (records and “collapse”)
When records form, the least-leaky, most-predictive update is entanglement-breaking / GKSL. “Collapse” is a priced edit, not a postulate.
Gauge + matter (signals and locks)
Local symmetries carry charges with fewest moving parts. Yukawa gauge-invariance + anomaly cancellation fix hypercharge up to scale; with a minimal CP-violation binder and strictly convex rep-costs, the unique replication is three families (strict gap to any other NN).
Big Bang as coherence nucleation
Expansion begins once the coherence reproduction numberR_(coh) > 1\mathcal{R}_{\rm coh}>1, producing FRW smoothing without an inflaton.
Dark matter as frozen-leakage domains
Sequestered, pointer-frozen regions behave as long-lived, gravitationally coupled matter.
Dark energy as persistent leakage freeze
Frozen leakage blocks appear as an effective cosmological constant with w~~-1w\approx -1.
Answering the biggest mysteries in physics
GR vs QM “don’t mesh.” They’re regimes of one selection: the cone + budgets pick GR for the backbone and GKSL for record-making. An “observer” is just a coherent pattern pushing a sub-coherent one past threshold.
Cosmology tensions (H_(0)_0, growth). “Laws” are local winners; finite budgets guarantee O(epsi^(eta))\mathcal{O}(\varepsilon^{\eta}) corrections that integrate over history. Fit once—across probes.
Constants are duals, not mysteries. Couplings are budget multipliers: measured values report how nature pays at that scale.
Principled BSM filter. New fields/interactions must lower total selection cost (increase Hold more than they add fuel/complexity/leakage). Otherwise: rejected on audit.
Predictions
Finite budgets imply small, structured departures from the GR/QM limits. A single suppression pair(epsi,eta)(\varepsilon,\eta) must jointly fit the signals below.
Gravitational-wave phasing — curvature/mass-scaled residuals on GR templates; no extra polarizations. Predicted scale: residuals ∼10^(-3)\sim 10^{-3} (LVK band).
Hawking flux — temperature fixed at T=kappa//2piT=\kappa/2\pi; amplitude suppressed. Predicted suppression:Delta F//F~~-(0.01+-0.004)Phi(omega//kappa)\Delta F/F \approx -(0.01 \pm 0.004) \Phi(\omega/\kappa).
Dark matter — effective equation of state w in[0,0.1]w \in [0,0.1]; CDM-like clustering in the cold limit.
Dark energy — pointer-frozen leakage =>\Rightarroww=-1w=-1 within current bounds.
One-knob test. A single (epsi,eta)(\varepsilon,\eta) must fit GW phasing + BH shadows + cosmology together. Success elevates the law from elegance to experiment; disagreement falsifies it.
Bottom line
Assume coherence, pay every bill, and the world compresses to the simplest pattern that still carries cause‑and‑effect—our spacetime with Einstein’s backbone, quantum’s editor, and gauge‑locked signals, plus tiny, priced cracks we can now measure.
Coherence Theory Paper V1.5
Abstract
We formalize coherence as the staying‑power of a pattern under admissible pokes, priced by three convex budgets—throughput, complexity, and leakage—selected by symmetry and locality. We prove budget minimal completeness (no fourth budget), poke‑ensemble robustness, and a non‑teleological variational principle via risk‑sensitive large deviations. Fast‑sector KKT on a C*‑quadratic fixes ℏ\hbar; slow‑sector Γ‑compactness recovers Einstein–Hilbert. Pointer alignment follows from a unitary‑orbit minimizer. Multi‑cone geometries pay a strict L¹ coherence penalty.
Global hypotheses
H0 (Spaces). Fast: separable Hilbert space H\mathcal H; states rho inT_(1)(H)\rho\in\mathfrak T_1(\mathcal H) (||*||_(1)\|\cdot\|_1). Pokes: CPTP maps (||*||_(diamond)\|\cdot\|_\diamond). Slow: fields in H_(loc)^(1)H^1_{\rm loc} modulo gauge/diffeo; Γ‑convergence frames locality. H1 (Budgets). Convex, l.s.c., coercive, invariance‑compatible, additive, monotone under coarse‑graining; slow Γ‑limit is second‑order local. H2 (Operational measurability). Budgets and CL\mathrm{CL} estimable from finite experiments; probabilities continuous in ||*||_(diamond)\|\cdot\|_\diamond. H3 (Coherence functional). l.s.c. in Phi\Phi; u.s.c. and concave in AA on budget sublevels. H4 (Causality/Locality for pokes). Single light‑cone; Γ‑locality (no super‑quadratic derivatives at selection scale). H5 (Leakage regularity). Transfer‑kernel lower hull continuous; strict hull convexity ⇒ unimodality; else a weak U‑shape suffices.
Patterns that keep working when poked are selected. Call this staying‑power coherence. Selection pays three prices: throughput (fuel/time/compute), complexity (moving parts/coordination), and leakage (unwanted emissions/crosstalk). Environments poke within causal limits. The winning scaffold maximizes predictive staying‑power minus these prices.
1.1.1 Assumptions Ledger (what we assume, where it is used)
We assume exactly three hygiene items; everything else is derived.
(A) Poke cone: a causal, Γ-local class \(\mathcal P\) of disturbances, closed under composition and mixing; product-topology continuity on finite windows. Used in: §1.2 (envelope), Ch. 2 (operational l.s.c.), Ch. 5 (directional envelope), Ch. 9 (microcausality).
(B) Budgets: three convex, l.s.c., coercive quadratics—throughputB_(th)B_{\rm th} (derivation-priced), complexityB_(cx)B_{\rm cx} (Ad-invariant Hilbertian), leakageB_(leak)B_{\rm leak} (Dirichlet-type on channels)—with norm-equivalent representatives and calibration stability. Used in: Ch. 2 (irreducible basis), Ch. 3 (fast sector/GKSL), Ch. 5 (multipliers), Ch. 7 (horizon).
(C) Spaces/topologies: quasi-local C* algebra for fast variables; cone-preserving Γ-compact slow sector (bounded geometry + gauge fixing). Used in: §1.2 (existence), Ch. 4 (Γ-limit ⇒ EH), Ch. 5 (first-variation convergence).
1.1.2 What we assume vs. what we derive vs. how to falsify
No fourth independent budget (irredundant 3-D span on feasible quotient); constants = multipliers
A fourth quadratic direction separates under the same symmetries/calibration
Fast sector
HS geometry on blocks; derivation price
ℏ=lambda_(th)^(-1)\hbar=\lambda_{\rm th}^{-1}; Heisenberg/GKSL with pointer basis (W-diagonalization)
Lab interferometer shows basis-invariant decoherence contrary to W-alignment
Slow sector
Γ-compact, cone-preserving class
Γ-limit ⇒ Einstein–Hilbert scaffold; coupled EH–YM + GKSL at stationarity
GW phasing residual slope departs from predicted envelope-multiplier law
Horizons
Same budgets; near-horizon cone
Amplitude suppression of Hawking flux at fixed temperature
Detect a temperature shift (leading order) instead of pure amplitude suppression
(Pointers to full proofs remain where those theorems live.)
1.2 The Coherence Law (auditable form)
Let A:=A_(fast)xxA_(slow)xxD\mathcal A:= \mathcal A_{\rm fast}\times\mathcal A_{\rm slow}\times \mathcal D be the product of the fast, slow, and discrete choices, equipped with the product topology (trace/diamond‑side for fast; cone‑preserving weak H^(2)H^2 for slow after gauge‑fixing; and the discrete topology on D\mathcal D). Let P\mathcal P be the admissible poke cone (causal, Γ‑local; closed under composition/mixing) and bar(P)\overline{\mathcal P} its diamond‑norm closure. Budgets B_(th),B_(cx),B_(leak)B_{\rm th},B_{\rm cx},B_{\rm leak} are convex, l.s.c., coercive and invariance‑compatible (Appendix A); binders B_(bind)^((j))B_{\rm bind}^{(j)} are l.s.c. and either indicator‑type or convex quadratics on submanifolds. The coherence functional CL\mathrm{CL} is operational and l.s.c. in pokes (Lemma II.2).
Operationalization (finite protocols) and risk-sensitive limit
Finite-protocol representation. There exists a family of experimentally finite protocols T inTT\in\mathscr T (finite POVMs plus bounded continuous post-processings) such that
CL(A,Phi)=s u p_(T inT)F_(T)(A,Phi),qquadF_(T)(A,Phi):=g_(T)(p_(T)(A,Phi)),\mathrm{CL}(A,\Phi)=\sup_{T\in\mathscr T} F_T(A,\Phi),\qquad
F_T(A,\Phi):=g_T\!\big(p_T(A,\Phi)\big),
with T|->F_(T)T\mapsto F_T continuous in the diamond norm. Hence Phi|->CL(A,Phi)\Phi\mapsto \mathrm{CL}(A,\Phi) is l.s.c. and auditable.
Risk-sensitive aggregator. For poke law Pi\Pi and beta > 0\beta>0,
CL_(beta)(A):=(1)/(beta)log E_(Phi∼Pi)exp (betaCL(A,Phi))↘i n f_(Phi in bar(P))CL(A,Phi)quad(beta rarr oo)\mathrm{CL}_\beta(A):=\frac{1}{\beta}\log\mathbb E_{\Phi\sim\Pi}\exp\big(\beta\,\mathrm{CL}(A,\Phi)\big)
\searrow \inf_{\Phi\in\overline{\mathcal P}}\mathrm{CL}(A,\Phi)\quad(\beta\to\infty)
(epi-convergence). Thus the worst-case envelope is the beta rarr oo\beta\to\infty risk limit—no teleology is assumed.
Global cross-domain KPI. We track the coherence number
the ratio of decoherence time (fast, pointer-aligned) to messenger time (slow, cone-propagating). chi\chi appears in both lab interferometers and near-horizon tiles and is linked to multipliers by the envelope identities (App. E.4).
Concrete coherence functionals (two exemplars)
We exhibit two computable coherence functionals CL\mathrm{CL} within the admissible class defined above. Both respect finite protocols and the poke cone, and both yield the same selection outputs up to a monotone transform (Appendix A1).
(A) Classical toy-world (cellular-automaton) CL.
State space: a finite grid Z_(n)^(2)\mathbb Z_n^2 with cell states S={0,1,2}S=\{0,1,2\} for empty/scaffold/messenger. A pattern AA is a seed a inS^(n xx n)a\in S^{n\times n} plus a local update rule R_( theta)R_\theta (finite-radius). A poke Phi\Phi is a Markovian disturbance with parameters (p_(noise),q_(adv))(p_{\rm noise},q_{\rm adv}) acting at each step for TT steps. A finite protocolT_(CA)T_{\rm CA} fixes thresholds (s_(min),tau_(hold),theta_(msg))(s_{\min},\tau_{\rm hold},\theta_{\rm msg}) and an evaluation schedule Ssub{1,dots,T}\mathcal S\subset\{1,\dots,T\}. Let XX be the trajectory under A,PhiA,\Phi.
Define three finite, observable functionals under T_(CA)T_{\rm CA}:
S_(A,Phi)^(T_(CA)):=P["there exists a 4-connected component of state "1" of size" >= s_(min)" that persists " >= tau_(hold)]S_{A,\Phi}^{T_{\rm CA}}:=\mathbb{P}[\text{there exists a 4-connected component of state }1\text{ of size}\ge s_{\min}\text{ that persists }\ge \tau_{\rm hold}].
M_(A,Phi)^(T_(CA)):=P[#{t inS:"messenger mass in core" >= m_(min)} >= theta_(msg)|S|]M_{A,\Phi}^{T_{\rm CA}}:=\mathbb{P}\big[ \#\{t\in\mathcal{S}:\text{messenger mass in core} \ge m_{\min}\}\ge \theta_{\rm msg}|\mathcal{S}|\big].
L_(A,Phi)^(T_(CA)):=E["leakage events in "X]//L_(cap)L_{A,\Phi}^{T_{\rm CA}}:=\mathbb{E}[\text{leakage events in }X]/L_{\rm cap} (dimensionless).
For weights u=(alpha,beta,gamma)inUsubDelta_(2)u=(\alpha,\beta,\gamma)\in\mathcal U\subset\Delta_2 (finite grid on the simplex), define the CA protocol score
This is finite-protocol, measurable, and l.s.c. in the diamond/trace product topology (Appendix A1, Lemma A1.1). Under randomized mixtures of patterns (Section 1.1.1), A|->CL_(CA)(A,Phi)A\mapsto \mathrm{CL}_{\rm CA}(A,\Phi) is concave (supremum of linear expectations over a finite family composed with an affine mixing).
(B) Quantum toy (binary channel reliability) CL.
Fix two finite-energy input states rho_(0),rho_(1)\rho_0,\rho_1 encoding “useful thing holds / fails.” For TT steps, a fast-sector pattern AA composed with poke Phi\Phi induces a CPTP map N_(A,Phi)\mathcal N_{A,\Phi}. Denote the outputs sigma _(i):=N_(A,Phi)(rho _(i))\sigma_i:=\mathcal N_{A,\Phi}(\rho_i). The optimal binary decision error for distinguishing sigma_(0)\sigma_0 vs. sigma_(1)\sigma_1 is Helstrom’s
with Delta:=rho_(0)-rho_(1)\Delta:=\rho_0-\rho_1. Then
CL_(Q)(A,Phi):=i n f_(Phi^(')in bar(P))F_(Q)(A,Phi^(')).\boxed{\ \mathrm{CL}_{\rm Q}(A,\Phi):=\inf_{\Phi'\in\overline{\mathcal P}}\ F_{\rm Q}(A,\Phi')\ }.
Continuity of N|->||N(Delta)||_(1)\mathcal N\mapsto \|\mathcal N(\Delta)\|_1 in diamond norm yields l.s.c. in (A,Phi)(A,\Phi); mixing AA gives concavity in the mixture (Appendix A1, Lemma A1.2). Choosing rho _(i)\rho_i aligned with the environmental weight WW connects this CL to the pointer-basis selection in Chapter 3.
Box 1.B — Robustness (no fine-tuning).
On any bounded window and admissible poke class, every CL in the family
{CL=s u p_(T inT)E[s_(T)(Z_(A,Phi))]:s_(T)" is a bounded, concave proper score on a finite observable "Z}\Big\{\ \mathrm{CL}=\sup_{T\in\mathscr T}\ \mathbb E\big[s_T(Z_{A,\Phi})\big]\ :\ s_T \text{ is a bounded, concave proper score on a finite observable }Z\ \Big\}
is equivalent up to an increasing bi-Lipschitz transform. Thus their maximizers under the same budgets coincide in the Gamma\Gamma-limit, and multipliers (e.g. ℏ=lambda_(th)^(-1)\hbar=\lambda_{\rm th}^{-1}) are invariant. (Proof: Appendix A1, Prop. A1.3.)
Well‑posedness & existence (direct method — full proofs)
Notation. Let C(A):= inf_{Phi in Pbar} CL(A,Phi) and J(A):= C(A) − lambda_th B_th(A) − lambda_cx B_cx(A) − lambda_leak B_leak(A) − sum_j mu_j B_bind^{(j)}(A). Fix c<∞ and denote
S_c := { A in A : lambda_th B_th(A)+lambda_cx B_cx(A)+lambda_leak B_leak(A)+sum_j mu_j B_bind^{(j)}(A) ≤ c }.
Prop. 1.2.1 (compact sublevels). Under H1, H4 and the cone‑preserving gauge‑fixed topology for the slow sector (Ch. 4), S_c is compact in the product topology fast × slow × discrete. Proof. By H1 each budget is lower semicontinuous and coercive in the stated product topology, so its sublevel sets are precompact. Finite intersections of precompact sets are precompact. Because all budgets and binders are l.s.c., S_c is closed; hence S_c is compact. The projection of S_c to the discrete factor is precompact; in a discrete space that implies finiteness, so only finitely many discrete labels occur on S_c. □
Prop. 1.2.2 (upper semicontinuity of A ↦ C(A)). Under H3, C is u.s.c. on A. Proof. Fix A and ε>0. Choose Phi_ε with C(A) ≥ CL(A,Phi_ε) − ε. For any net A_α→A, u.s.c. of A ↦ CL(A,Phi_ε) on budget sublevels (H3) gives limsup_α CL(A_α,Phi_ε) ≤ CL(A,Phi_ε). Hence limsup_α C(A_α) ≤ C(A)+ε. Let ε↓0. □
Thm. 1.2.3 (existence of maximizers). The set Argmax_{A in A} J(A) is nonempty. Proof. By construction CL is bounded above (built from probabilities; H2), so C(A) ≤ M for some finite M (normalize M=1 w.l.o.g.). Let M*:=sup_A J(A) and pick a maximizing net A_α with J(A_α)→M*. Then for some finite c the budget sum at A_α is ≤ c for all large α, so A_α∈S_c. By Prop. 1.2.1, S_c is compact; pass to a convergent subnet A_{α_k}→A*. By Prop. 1.2.2 and l.s.c. of budgets, J is u.s.c. on S_c, so M* = limsup_k J(A_{α_k}) ≤ J(A*) ≤ M*. Thus A* attains the supremum. □
Cor. 1.2.4 (tightness of discrete choices). Any maximizing net is eventually supported on a finite subset of the discrete menu; the discrete factor does not spoil compactness. □
Lemma 1.2.5 (inner attainment or minimizing net). For fixed A, the map Phi ↦ CL(A,Phi) is l.s.c. on the closed set Pbar (Lemma II.2). Hence: (i) if some sublevel {Phi : CL(A,Phi) ≤ t} is diamond‑precompact for t>C(A), then there exists Phi*(A) ∈ Pbar with CL(A,Phi*)=C(A); (ii) in general, there exists a minimizing net Phi_β(A) with CL(A,Phi_β) ↓ C(A). Proof. (i) l.s.c. + compact sublevel ⇒ minimum attained. (ii) pick a directed family of ε‑minimizers with ε ↓ 0. □
Thm. 1.2.6 (Danskin–Valadier envelope). Assume for each Phi the directional derivative D^+_A CL(A,Phi;h) exists on S_c. Then for every A∈S_c and direction h,
D^+ C(A;h) = inf{ D^+_A CL(A,Phi;h) : Phi ∈ Argmin(A) },
with the right side understood as the infimum over cluster points of minimizing nets when Argmin(A)=∅. If Argmin(A) is nonempty and compact, the infimum is attained. Proof. Standard marginal‑function formula (Rockafellar–Wets, Variational Analysis, Thm. 10.31) using u.s.c. in A and l.s.c. in Phi; extend to noncompact Phi by epigraphical limits and minimizing nets (Valadier 1973). □
Remark (measurable selections). On any Polish slice of S_c the multifunction A↦Argmin(A) admits Borel ε‑selections (Kuratowski–Ryll‑Nardzewski) when nonempty/closed; when empty use the minimizing‑net construction to define ε‑selectors, which suffices for Appendix J’s estimation schemes. Lemma 1.2.A (precompact sublevels & u.s.c.). For each c < ooc<\infty, the sublevel set S_(c):={A inA:lambda_(th)B_(th)(A)+lambda_(cx)B_(cx)(A)+lambda_(leak)B_(leak)(A)+sum _(j)mu _(j)B_(bind)^((j))(A) <= c}\mathsf S_c:=\Big\{A\in\mathcal A:\ \lambda_{\rm th} B_{\rm th}(A)+\lambda_{\rm cx} B_{\rm cx}(A)+\lambda_{\rm leak} B_{\rm leak}(A)+\textstyle\sum_j\mu_j B_{\rm bind}^{(j)}(A)\le c\Big\}
is precompact in the product topology, and the map A|->C(A)=i n f_(Phi in bar(P))CL(A,Phi)A\mapsto \mathcal C(A)=\inf_{\Phi\in\overline{\mathcal P}}\mathrm{CL}(A,\Phi) is upper semicontinuous on S_(c)\mathsf S_c. Sketch. Precompactness: fast‑sector coercivity (Appendix A) gives compactness of generator/state sublevels in the trace/diamond product; slow‑sector Γ‑compactness (Ch. 4) gives weak H^(2)H^2 precompactness under cone‑preserving bounds; D\mathcal D is either finite or handled by the discrete clause below. Upper semicontinuity: Phi|->CL(A,Phi)\Phi\mapsto\mathrm{CL}(A,\Phi) is l.s.c. in ||*||_(diamond)\|\cdot\|_\diamond (Lemma II.2), and A|->CL(A,Phi)A\mapsto\mathrm{CL}(A,\Phi) is u.s.c. on budget sublevels (H3); infima of such families preserve u.s.c. on S_(c)\mathsf S_c.
Lemma 1.2.B (outer maximizer exists). The objective is u.s.c. on each S_(c)\mathsf S_c and S_(c)\mathsf S_c is precompact; hence the outer arg max\arg\max is non‑empty. Any maximizing sequence has a convergent subsequence with a maximizer in the limit.
Discrete choices. Either (a) work with a compactified bar(D)\overline{\mathcal D} (e.g., finite menu or one‑point compactification that carries no advantage by coercivity), or (b) impose a finite‑improvement property: on each S_(c)\mathsf S_c only finitely many discrete flips can strictly improve the objective (holds when each flip increases at least one active budget by a fixed epsi > 0\varepsilon>0). This ensures attainment over D\mathcal D.
Inner attainment & envelope
Lemma 1.2.C (attainment/minimizing net for inner infimum). For fixed AA, the map Phi|->CL(A,Phi)\Phi\mapsto \mathrm{CL}(A,\Phi) is l.s.c. on the closed set bar(P)\overline{\mathcal P}. Thus the infimum is attained whenever the relevant slice of bar(P)\overline{\mathcal P} is diamond‑precompact; otherwise there exists a minimizing netPhi _(alpha)(A)\Phi_\alpha(A) with CL(A,Phi _(alpha))darrC(A)\mathrm{CL}(A,\Phi_\alpha)\downarrow \mathcal C(A). In either case, the Danskin–Valadier envelope applies: for any direction delta A\delta A, D^(+)C(A;delta A)=i n f{del _(A)CL(A,Phi)[delta A]:Phi in Argmin(A)}D^+\mathcal C(A;\delta A)=\inf\{\partial_A\mathrm{CL}(A,\Phi)[\delta A]:\ \Phi\in \operatorname{Argmin}(A)\}
with the right derivative taken on S_(c)\mathsf S_c. This is the version used later for KKT and constants‑as‑multipliers.
Notes. (i) No teleology (risk‑sensitive beta rarr oo\beta\to\infty limit) is shown in Ch. 2. (ii) Canonical budgets only is Theorem I.1. They are referenced here but not assumed—existence follows from Lemmas 1.2.A–C plus H0–H5.
Box 1.A — Three-budget sufficiency (no fourth direction).
Under the admissible symmetries and calibration stability, the admissible budgets span an irreducible 3-dimensional cone on the feasible quotient. A fourth independent quadratic is excluded by separation (Hahn–Banach) at fixed calibration. (Proofs in Ch. 2: Lemma I.5, Theorem I.1.)
1.3 Fast and slow sectors (what is being selected)
Fast: quantum‑like sector on a separable Hilbert space; states evolve by GKSL; in the zero‑leakage limit the evolution is unitary. The throughput budget is a C*‑compatible quadratic of the derivation delta _(H)(A)=i[H,A]\delta_H(A)=i[H,A]. KKT/Riesz in Chapter 3 fixes ℏ=lambda_(th)^(-1)\hbar=\lambda_{\rm th}^{-1}.
Slow: geometry and gauge scaffold selected by Γ‑limits under cone‑preserving, gauge‑fixed topologies; equi‑coercivity and Γ‑compactness proved in Chapter 4.
Global convention. Throughout this chapter all quadratic forms, operator adjoints, and norms on blocks are taken with respect to the unweighted normalized Hilbert–Schmidt (HS) geometry
Superoperator norms |*|_(HS rarr HS)|\cdot|_{HS \to HS}, cb-norms, and all adjoints ^(!){}^{!} are computed in this geometry. This fixes inner-product consistency across Lemmas I.1–I.3 and the budgets.
Why only these three? On each finite block, admissible budget functionals are support functions of convex, symmetry-invariant sets. Ad-invariance and additivity force the derivation-priced throughput, an Ad-invariant Hilbertian complexity, and a Dirichlet-type leakage as a complete basis. Passing to the feasible quotient and using Hahn–Banach separation at fixed calibration excludes any fourth independent quadratic. Null-budget directions are modded out; norm-equivalent representatives preserve the multipliers.
Definition I.0 (Admissible budget class — finalized)
Let omega\omega be a faithful normal state with GNS triple (pi _(omega),H_(omega),Omega _(omega))(\pi_\omega,\mathcal{H}_\omega,\Omega_\omega) and quasi-local C*-algebra A= bar(uuu _(Lambda)A_(Lambda))\mathcal{A}=\overline{\bigcup_\Lambda \mathcal{A}_\Lambda} built from finite blocks Lambda\Lambda with matrix algebras M_(d_( Lambda))M_{d_\Lambda}. Equip each M_(d_( Lambda))M_{d_\Lambda} and CB(M_(d_( Lambda)))\mathrm{CB}(M_{d_\Lambda}) with their canonical operator-space structures and the normalized HS geometry above.
A budget is a map B:Ararr[0,oo]\mathcal{B}:\mathcal{A}\to[0,\infty] acting on scaffolds A=(A_(fast),A_(med),A_(leak),A_(slow),A_(disc))A=(A_{\rm fast},A_{\rm med},A_{\rm leak},A_{\rm slow},A_{\rm disc}) and satisfying:
(H1) Convexity, l.s.c., coercivity.B\mathcal{B} is convex, lower semicontinuous for the product of blockwise HS topologies, and its sublevel sets intersected with the feasible class are relatively compact in the cone-preserving topology (Ch. 4).
Invariance.
(Fast/med)B\mathcal{B} is invariant under block-local unitaries UU (Ad-invariance) and relabeling symmetries.
(Leak) It is invariant under Kraus-index mixing by V in U(m)V\in U(m) and equivariant under system unitaries:B_(leak)^(Lambda)({UL_(j)U^(†)};UWU^(†))=B_(leak)^(Lambda)({L_(j)};W).\mathcal{B}_{\rm leak}^\Lambda(\{UL_jU^\dagger\};UWU^\dagger)=\mathcal{B}_{\rm leak}^\Lambda(\{L_j\};W).
Second-order locality. On each block Lambda\Lambda, the fast/mediator/leakage parts restrict to quadratic forms computed from the HS pairing above; the global budget is the inductive-limit supremum of block quadratics.
Functorial ampliation. Whenever a cb-norm (or a completely Hilbertian norm) appears, it is complete: |id_(k)oxJ|=|J||{\rm id}_k\otimes \mathcal{J}|=|\mathcal{J}| for all k inNk\in\mathbb{N}.
(H0.loc) Bounded local dimension. There exists d_(site) < ood_{\rm site}<\infty and a locality radius r inNr\in\mathbb{N} such that any JinLoc_(r)(Lambda)\mathcal{J}\in\mathsf{Loc}_r(\Lambda) acts nontrivially on at most C(r)C(r) sites, whence its support algebra is ~=M_(d_(loc))\cong M_{d_{\rm loc}} with d_(loc):=d_(site)^(C(r))d_{\rm loc}:=d_{\rm site}^{C(r)} independent of Lambda\Lambda.
(H5) Complete CP-monotonicity for leakage (sub-Markov in $W$). For any admissible block-local CP maps $\Phi_{\rm pre},\Phi_{\rm post}$ whose HS-adjoints satisfy
for a fixed positive quadratic form $\mathfrak q_\Lambda$ independent of $W$ (calibration fixes its scale on rank-one $P_a$).
Remark (Dirichlet linearity). Axiom (H5.lin) is the Markov/Dirichlet linearity postulate for leakage: the leakage quadratic depends linearly on the pointer weight $W$ and is orthogonally additive along its spectral decomposition. It is satisfied by the canonical GKSL-weighted form and is equivalent, in finite dimension, to requiring that the leakage quadratic be a left Dirichlet form in $W$ with no weight-independent (bare) component.
Denote by B\mathfrak{B} the cone of budgets satisfying 1–5, and by bar(F)\overline{\mathcal{F}} the feasible closure (Ch. 1.2).
On a block Lambda\Lambda, any Ad-invariant convex quadratic Q_( Lambda)Q_\Lambda of the inner derivation delta_(H)^( Lambda)=[H,*]\delta_H^\Lambda=[H,\cdot] (viewed as a superoperator on (M_(d_( Lambda)),(:*,*:)_(2;Lambda))(M_{d_\Lambda},\langle\cdot,\cdot\rangle_{2;\Lambda})) is a constant multiple of B_(th)^(Lambda)(H)=(1)/(2)||delta_(H)^( Lambda)||_(HS rarr HS)^(2)=(1)/(2)(1)/(d_( Lambda))Tr_(HS)((delta_(H)^( Lambda))^(†)delta_(H)^( Lambda)).\mathcal{B}^{\Lambda}_{\rm th}(H)=\tfrac{1}{2}\,\|\delta_H^\Lambda\|_{HS\to HS}^2=\tfrac{1}{2}\,\frac{1}{d_\Lambda}\operatorname{Tr}_{HS}\!\big((\delta_H^\Lambda)^\dagger\delta_H^\Lambda\big).
Calibrating on a two-site reference fixes the constants uniformly, yielding the inductive-limit budget
B_(th)(H)=s u p _(Lambda)B_(th)^(Lambda)(H).\boxed{\ \mathcal{B}_{\rm th}(H)=\sup_\Lambda \mathcal{B}^{\Lambda}_{\rm th}(H)\ }. Proof. The adjoint representation of U(d_( Lambda))U(d_\Lambda) on su(d_( Lambda))\mathfrak{su}(d_\Lambda) is irreducible; by Schur, any Ad-invariant bilinear form is a scalar multiple of the Killing form. Passing to superoperators delta _(H)\delta_H preserves Ad-covariance; the unique (up to scale) Ad-invariant quadratic is the HS operator-norm square shown. ◻\square
Lemma I.2 (Mediator/complexity via a completely bounded Hilbertian norm — airtight)
Let Loc_(r)(Lambda)subCB(M_(d_( Lambda)))\mathsf{Loc}_r(\Lambda)\subset \mathrm{CB}(M_{d_\Lambda}) denote block-local maps with radius <= r\le r (Def. I.0(5)). Suppose Q_( Lambda)Q_\Lambda is a convex quadratic on Loc_(r)(Lambda)\mathsf{Loc}_r(\Lambda) that is (i) unitarily covariant, (ii) functorially ampliation-stable, (iii) second-order local and cb-continuous. Then there exist constants c_(1),c_(2)in(0,oo)c_1,c_2\in(0,\infty) depending only on (r,d_(site))(r,d_{\rm site}) such that for every Lambda\Lambda and JinLoc_(r)(Lambda)\mathcal{J}\in\mathsf{Loc}_r(\Lambda),
Q_( Lambda)(J)≃||J||_(h,2)^(2):=i n f{sum_(k)||a_(k)||_(2;Lambda)^(2)||b_(k)||_(2;Lambda)^(2):J(X)=sum _(k)a_(k)Xb_(k)},Q_\Lambda(\mathcal J)\ \simeq\ \|\mathcal J\|_{h,2}^2
:=\inf\Big\{\sum_{k}\|a_k\|_{2;\Lambda}^2\,\|b_k\|_{2;\Lambda}^2:\ \mathcal J(X)=\sum_k a_k X b_k\Big\},
with equivalence constants depending only on (r,d_(site))(r,d_{\rm site}). Consequently, a block complexity budget can be taken—up to those constants—as
where the weights kappa _(alpha)\kappa_\alpha encode overlap counts from locality. The value is decomposition-independent; the inductive-limit budget is B_(cx)=s u p _(Lambda)B_(cx)^(Lambda)\mathcal{B}_{\rm cx}=\sup_\Lambda \mathcal{B}^{\Lambda}_{\rm cx}.
Proof.
(A) Dimension-free reduction. By (H0.loc), each JinLoc_(r)(Lambda)\mathcal{J}\in\mathsf{Loc}_r(\Lambda) factors through M_(d_(loc))M_{d_{\rm loc}} with d_(loc)=d_(site)^(C(r))d_{\rm loc}=d_{\rm site}^{C(r)} independent of Lambda\Lambda.
(B) Haagerup–Hilbertian control. On M_(d_(loc))M_{d_{\rm loc}}, functorial ampliation and unitary covariance imply Q_( Lambda)Q_\Lambda is completely Hilbertian; operator-space duality identifies it (up to constants depending only on d_(loc)d_{\rm loc}) with the Haagerup-Hilbertian seminorm |*|_(h,2)|\cdot|_{h,2}, cb-equivalent in fixed finite dimension (standard Haagerup–Pisier cb≃Hilbertian equivalence on M_(d_(loc))M_{d_{\rm loc}}).
(C) Decomposition independence. In finite dimension the Haagerup projective cone is closed; strong duality equates the projective infimum with Q_( Lambda)Q_\Lambda. Pull back along the locality factorization. ◻\square
Lemma I.3 (Leakage factorization as a completely Dirichlet form — airtight)
Let L_(leak)^(Lambda)\mathcal{L}_{\rm leak}^\Lambda assign to a Kraus list {L_(j)}_(j=1)^(m)subM_(d_( Lambda))\{L_j\}_{j=1}^m\subset M_{d_\Lambda} a convex quadratic satisfying: system-unitary equivariance in WW (Def. I.0(2)), Kraus-mixing invariance, (H5) complete CP-monotonicity with Phi^(!)(W) <= W\Phi^{!}(W)\le W, and (H5.lin) weight linearity. Then there exists a positive affiliated weight W>-0W\succ0 (unique up to conjugation and equivalence on bar(F)\overline{\mathcal{F}}) such that
B_(leak)^(Lambda)({L_(j)};W)=sum_(j=1)^(m)||W^(1//2)L_(j)||_(2;Lambda)^(2)=sum_(j=1)^(m)(1)/(d_( Lambda))tr(L_(j)^(†)WL_(j)).\boxed{\ \mathcal B^{\Lambda}_{\rm leak}(\{L_j\};W)=\sum_{j=1}^m \|W^{1/2}L_j\|_{2;\Lambda}^2
=\sum_{j=1}^m\frac{1}{d_\Lambda}\operatorname{tr}\!\big(L_j^\dagger W L_j\big)\ .}
Proof. Work on a fixed block, drop Lambda\Lambda.
(1) Kernel reduction. Kraus-mixing invariance gives Q({L_(j)})=sum _(j)(:L_(j),T_(W)L_(j):)_(2)Q(\{L_j\})=\sum_j \langle L_j,T_W L_j\rangle_{2} for some positive linear T_(W)T_W on M_(d)M_d. System-unitary covariance implies T_(UWU^(†))=Ad_(U)@T_(W)@Ad_(U^(†))T_{U W U^\dagger}=\mathrm{Ad}_U\circ T_W\circ \mathrm{Ad}_{U^\dagger}.
(2) Spectral diagonalization. Let W=sum _(a)w_(a)P_(a)W=\sum_a w_a P_a. Using pre/post pinching that fix WW (so (Pi^(L))^(!)(W)=(Pi^(R))^(!)(W)=W(\Pi^{\rm L})^{!}(W)=(\Pi^{\rm R})^{!}(W)=W and (H5) applies), one gets
Q(L)=sum_(a,b)Q(P_(a)LP_(b)),Q(L)=\sum_{a,b}Q(P_a L P_b),
hence T_(W)T_W commutes with L_(P_(a))L_{P_a} and R_(P_(b))R_{P_b} and takes the form
Comparing with the decomposition above and varying {w_(a)}\{w_a\} yields that t_(ab)(W)t_{ab}(W) depends only on the left eigenvalue and linearly: t_(ab)(W)=alpha(w_(a))t_{ab}(W)=\alpha(w_a) with alpha\alpha positive linear and independent of bb. Consequently,
No right term and no bare (weight-independent) term are admissible under (H5.lin).
(4) Identify A(W)=kappa WA(W)=\kappa W. Covariance within multiplicity spaces and homogeneity force alpha(w)=kappa w\alpha(w)=\kappa w; hence A(W)=kappa WA(W)=\kappa W and
Calibrate on a two-site reference to fix kappa=1\kappa=1. Uniqueness up to conjugation/equivalence follows from faithfulness and polarization. ◻\square
with the product HS topology. Let B_(Lambda)\mathcal{B}_\Lambda be the restriction of BinB\mathcal{B}\in\mathfrak{B} to V_(Lambda)\mathcal{V}_\Lambda. Then:
epi(B_(Lambda))\operatorname{epi}(\mathcal{B}_\Lambda) is a closed convex cone; by Fenchel–Moreau in finite dimension,
B_(Lambda)(X)=s u p_(S inS_(Lambda))(:S,X:).\mathcal{B}_\Lambda(X)=\sup_{S\in\mathscr{S}_\Lambda}\ \langle S,X\rangle.
Averaging SS over the compact symmetry groups and admissible coarse-grains is continuous and value-preserving; the averaged admissible observables form a compact set.
By Lemmas I.1–I.3, the invariant quadratic subspace on V_(Lambda)\mathcal{V}_\Lambda is three-dimensional, generated by (B_(th)^(Lambda),B_(cx)^(Lambda),B_(leak)^(Lambda))(\mathcal{B}_{\rm th}^\Lambda,\mathcal{B}_{\rm cx}^\Lambda,\mathcal{B}_{\rm leak}^\Lambda). Thus each block support functional reduces to a triple
Lemma I.5 (Hahn–Banach separation ⇒ three-dimensionality on the feasible quotient
Let Q\mathcal{Q} be the quotient of the linear span of {B_(Lambda):Lambda}\{\mathcal{B}_\Lambda:\Lambda\} by the subspace of null budgets{N:N|_( bar(F))=0}\{\mathcal{N}:\mathcal{N}|_{\overline{\mathcal{F}}}=0\}. The observable cone identified in Prop. I.4 is closed and equals cone{s_(th),s_(cx),s_(leak)}\operatorname{cone}\{s_{\rm th},s_{\rm cx},s_{\rm leak}\}. A putative fourth independent admissible budget would be separated by a continuous observable, contradicting Prop. I.4(3). Hence
with C(r)C(r) counting finite overlaps. Two-site calibration fixes a common scale; the proportionality constants coincide across blocks. Therefore the inductive-limit budgets
B_(th)=s u p _(Lambda)B_(th)^(Lambda),qquadB_(cx)=s u p _(Lambda)B_(cx)^(Lambda),qquadB_(leak)=s u p _(Lambda)B_(leak)^(Lambda)\mathcal B_{\rm th}=\sup_\Lambda \mathcal B_{\rm th}^\Lambda,\qquad
\mathcal B_{\rm cx}=\sup_\Lambda \mathcal B_{\rm cx}^\Lambda,\qquad
\mathcal B_{\rm leak}=\sup_\Lambda \mathcal B_{\rm leak}^\Lambda
are well-defined, l.s.c., and coercive with block-independent constants.
Lemma I.6 (Null budgets form a closed subspace)
Let N:={N:N|_( bar(F))=0}\mathsf{N}:=\{\mathcal{N}:\mathcal{N}|_{\overline{\mathcal{F}}}=0\}. Then N\mathsf{N} is a closed linear subspace for the inductive-limit topology.
Proof. If N_(k)rarrN\mathcal{N}_k\to\mathcal{N} and each vanishes on bar(F)\overline{\mathcal{F}}, then for any A in bar(F)A\in\overline{\mathcal{F}} and large enough Lambda\Lambda, A|_(Lambda)A|_\Lambda is feasible and N_(k)(A)rarrN(A)\mathcal{N}_k(A)\to \mathcal{N}(A) by l.s.c.; hence N(A)=0\mathcal{N}(A)=0. ◻\square
Theorem I.1 (Irreducible basis of admissible budgets — airtight)
Under Definition I.0 (including (H5.lin)), the Consistency Lemma, and Lemma I.6, any BinB\mathcal{B}\in\mathfrak{B} admits the decomposition
No fourth independent budget satisfying 1–7 exists.
Proof. (i) Blockwise representation. Prop. I.4 expresses B_(Lambda)\mathcal{B}_\Lambda as a support function over cone{s_(th),s_(cx),s_(leak)}\operatorname{cone}\{s_{\rm th},s_{\rm cx},s_{\rm leak}\}, hence a conic combination of (B_(th)^(Lambda),B_(cx)^(Lambda),B_(leak)^(Lambda))(\mathcal{B}_{\rm th}^\Lambda,\mathcal{B}_{\rm cx}^\Lambda,\mathcal{B}_{\rm leak}^\Lambda) modulo nulls.
(ii) Inductive limit. Two-sided bounds and equi-coercivity allow a diagonal selection so that the supremum passes to the limit.
(iii) Exclusion of a fourth direction. Lemma I.5 rules it out on the feasible quotient.
(a) Calibration. After fixing a canonical two-site calibration, replacing any canonical quadratic by a blockwise norm-equivalent representative (constants depending only on (r,d_(site))(r,d_{\rm site})) rescales the associated alpha\alpha by the fixed calibration factor.
(b) KKT multiplier equality under regularity. If the optimization on bar(F)\overline{\mathcal{F}} enjoys Slater interior and strong convexity on feasible slices (so KKT multipliers are unique and stable), then any calibrated norm-equivalent replacement of a canonical quadratic leaves the Lagrange multipliers identical. Without strong convexity, multipliers are preserved up to the fixed calibration constants in (a). ◻\square
2.2 — Poke ensemble robustness (Block II)
Definition II.1 (Admissible poke cone)
P\mathcal{P} is the smallest set of CPTP maps on T_(1)(H)\mathfrak{T}_1(\mathcal{H}) that is: (i) causal (single cone), (ii) Gamma\Gamma-local, (iii) closed under convex mixing and composition, and (iv) contains neighborhoods of id\mathrm{id} and of mixing channels at all allowed scales. Let bar(P)\overline{\mathcal{P}} be its diamond-norm closure.
Lemma II.2 (Operational l.s.c.)
Under H2–H3, for fixed AA the map Phi|->CL(A,Phi)\Phi\mapsto\mathrm{CL}(A,\Phi) is lower semicontinuous in the diamond norm |*|_(diamond)|\cdot|_\diamond.
Proof (operational representation ⇒ l.s.c.). By H2 there exists a directed family T\mathscr{T} of finite experimental protocolsTT (finitely many channel uses, interleaved with fixed CPTP pre/post-processing and POVMs) and bounded continuous post-processings g_(T)g_T such that
CL(A,Phi)=s u p_(T inT)F_(T)(A,Phi),qquadF_(T)(A,Phi):=g_(T)(p_(T)(A,Phi)),\mathrm{CL}(A,\Phi)=\sup_{T\in\mathscr{T}} F_T(A,\Phi),\qquad F_T(A,\Phi):=g_T\!\big(p_T(A,\Phi)\big),
where p_(T)(A,Phi)p_T(A,\Phi) is the finite outcome-probability vector generated by TT.
Fix AA and TT using at most NN calls to Phi\Phi. A telescoping/adaptivity bound gives
hence (POVM contractivity) |p_(T)(A,Phi)-p_(T)(A,Psi)|_(1) <= N|Phi-Psi|_(diamond)|p_T(A,\Phi)-p_T(A,\Psi)|_1\le N|\Phi-\Psi|_\diamond. With g_(T)g_T continuous on the simplex, F_(T)(A,*)F_T(A,\cdot) is continuous in |*|_(diamond)|\cdot|_\diamond. A pointwise supremum of continuous functions is l.s.c.; therefore Phi|->CL(A,Phi)\Phi\mapsto\mathrm{CL}(A,\Phi) is l.s.c. ◻\square
Theorem II.3 (Equivalence of ensembles / robustness)
For every A inAA\in\mathcal{A},
i n f_(Phi inP)CL(A,Phi)=i n f_(Phi in bar(P))CL(A,Phi).\inf_{\Phi\in \mathcal{P}}\mathrm{CL}(A,\Phi)=\inf_{\Phi\in \overline{\mathcal{P}}}\mathrm{CL}(A,\Phi).
Proof. Lower semicontinuity (Lemma II.2) plus "infimum over a set equals infimum over its closure." ◻\square
Corollary II.4 (Leakage envelope attainment)
With (H5), the spectral transfer envelope w^(**)(nu)w^*(\nu)attains a minimum on bar(P)\overline{\mathcal{P}} (not necessarily unique). ◻\square
2.3 — Selection mechanics & large deviations (Block III)
Let pokes Phi _(t)∼^(i.i.d.)Pi\Phi_t\stackrel{\rm i.i.d.}{\sim}\Pi with support dense in bar(P)\overline{\mathcal{P}}. Define the risk-sensitive score (beta > 0\beta>0):
LD hypotheses. (LD1) Finite log-MGF on budget sublevels, uniformly on compacta. (LD2) Coercivity (H1) ⇒ exponential tightness. (LD3) A|->CL_(beta)(A)A\mapsto\mathrm{CL}_\beta(A) u.s.c. on sublevels.
Lemma III.1 (Risk-sensitive ⇒ worst-case)
lim_(beta rarr oo)CL_(beta)(A)=i n f_(Phi in bar(P))CL(A,Phi).\lim_{\beta\to\infty}\mathrm{CL}_\beta(A)=\inf_{\Phi\in\overline{\mathcal{P}}}\mathrm{CL}(A,\Phi).
Proof. Varadhan's lemma for log E[e^(-betaCL)]\log\mathbb{E}[e^{-\beta\,\mathrm{CL}}] yields the convex conjugate; as beta rarr oo\beta\to\infty, entropy regularization vanishes and the essential infimum remains. Density plus l.s.c. (Lemma II.2) give equality on bar(P)\overline{\mathcal{P}}. ◻\square
lim_(beta rarr oo)lim_(t rarr oo)(1)/(t)log Z_(t)^((beta))=s u p_(A inA)[i n f_(Phi in bar(P))CL(A,Phi)-alpha_(th)B_(th)(A)-alpha_(cx)B_(cx)(A)-alpha_(leak)B_(leak)(A)].\lim_{\beta\to\infty}\lim_{t\to\infty}\frac1t\log Z_t^{(\beta)}
= \sup_{A\in\mathcal A}\Big[\inf_{\Phi\in\overline{\mathcal P}}\mathrm{CL}(A,\Phi)\ -\ \alpha_{\rm th}B_{\rm th}(A)\ -\ \alpha_{\rm cx}B_{\rm cx}(A)\ -\ \alpha_{\rm leak}B_{\rm leak}(A)\Big].
Proof. Laplace principle under (LD2–LD3) gives lim_(t rarr oo)t^(-1)log Z_(t)^((beta))=s u p _(A)[CL_(beta)-sumalpha_(∙)B_(∙)]\lim_{t\to\infty}t^{-1}\log Z_t^{(\beta)}=\sup_A[\mathrm{CL}_\beta-\sum\alpha_\bullet B_\bullet]. Apply Lemma III.1 and epi-convergence (monotone in beta\beta). ◻\square
2.4 — Envelope identities (constants as multipliers)
Let Phi(tau)\Phi(\tau) be the optimal value with budget allowances tau=(tau_(th),tau_(cx),tau_(leak))\tau=(\tau_{\rm th},\tau_{\rm cx},\tau_{\rm leak}). Under H1 and Slater interior, Phi\Phi is convex in tau\tau and for a.e. tau\tau,
are the Lagrange multipliers (couplings/constants). Strict convexity along active directions ⇒ uniqueness; else an epi-small strictly convex regularizer from (H5) removes ties without new parameters.
Outcome of Blocks I–III. With (H5.lin) added, the admissible-budget cone is exactly three-dimensional up to nulls, generated by (B_(th),B_(cx),B_(leak))(B_{\rm th},B_{\rm cx},B_{\rm leak}) in the normalized HS geometry; leakage is a completely Dirichlet quadratic sum _(j)|W^(1//2)L_(j)|_(2)^(2)\sum_j |W^{1/2}L_j|_2^2 (calibrated); the poke-ensemble choice is robust to taking closures; and the selection principle is a worst-case limit of risk-sensitive growth with constants given by envelope multipliers.
Not fine-tuned. Any CL chosen from the bounded-concave proper-score family (Appendix A1) is equivalent up to an increasing transform; budget selection and multipliers are invariant.
Ablation note (empirical refuter). If we drop linear-response regularity (H5.lin) or relax Ad-invariance, mixed bimodule terms re-enter the admissible class and a fourth quadratic does separate on the observable cone, violating Lemma I.5. This yields a concrete falsifier: observe persistence of a fourth direction under the same calibration and the three-budget claim fails.
Chapter 3 — Fast Sector: Zero-Leakage ⇒ Unitary; ℏ\hbar as Throughput Dual; Pointer Basis
Scope. We make the budgets C*-compatible, correct the stationarity/dynamics bridge by pricing motion along the orbit (not the superoperator norm), derive A^(˙)=(i)/(ℏ)[H^(**),A]\dot{A}=\tfrac{i}{\hbar}[H^*,A] from a unitary-path variational principle (with a fully explicit variation calculus) and an equivalent pointwise/Pontryagin control derivation, prove multiplier stability under blockwise epi/Mosco limits and Ad-invariant rescalings, and give a unitary-orbit minimizer for leakage (pointer basis).
Bridge (what Chapter 3 actually fixes). At KKT stationarity on the unitary manifold with the normalized HS metric, the throughput multiplier identifies
With leakage re-enabled and the Dirichlet budget B_(leak)B_{\rm leak}, GKSL generators minimize leakage by W-alignment, selecting the pointer basis via co-diagonalization with the environmental weight WW.
How CL_(Q)\mathrm{CL}_{\rm Q} selects the pointer basis.
For fixed rho_(0),rho_(1)\rho_0,\rho_1 and environmental weight WW, minimizing leakage (Dirichlet budget) at fixed ||N_(A,Phi)(Delta)||_(1)\|\mathcal{N}_{A,\Phi}(\Delta)\|_1 forces co-diagonalization with WW; hence GKSL generators that align with the WW-eigenbasis are optimal. This is the pointer basis selection seen in §3.4; the argument is entirely in terms of the concrete CL_(Q)\mathrm{CL}_{\rm Q}.
3.1 — H0′: Spaces, norms, and budgets (C*-compatible)
Quasi-local algebra & GNS. Let A= bar(uuu _(Lambda)A_(Lambda))\mathcal{A}=\overline{\bigcup_\Lambda\mathcal{A}_\Lambda} be a quasi-local C*-algebra with faithful state omega\omega and GNS triple (pi _(omega),H_(omega),Omega _(omega))(\pi_\omega,\mathcal{H}_\omega,\Omega_\omega). Identify A inAA\in\mathcal{A} with pi _(omega)(A)subB(H_(omega))\pi_\omega(A)\subset\mathcal{B}(\mathcal{H}_\omega).
Common core & derivation. Fix a dense invariant core DsubH_(omega)\mathcal{D}\subset\mathcal{H}_\omega common to all unbounded generators considered. For (essentially) self-adjoint HH with DsubDom(H)\mathcal{D}\subset\mathrm{Dom}(H), define on the local ***-algebra A_(loc)\mathcal{A}_{\rm loc}:
delta _(H)(A):=i[H,A],qquad A inA_(loc).\delta_H(A):=i[H,A],\qquad A\in\mathcal{A}_{\rm loc}.
Assume delta _(H)\delta_H is closable there; write its closure again as delta _(H)\delta_H.
Normalized HS geometry on finite blocks. On each finite Lambda\Lambda (matrix algebra M_(d_( Lambda))M_{d_\Lambda}),
Complexity budget (local CP decompositions). For local CP pieces {J_(alpha)}\{\mathcal{J}_\alpha\} with cb-norms |*|_(cb)|\cdot|_{cb} and scale weights kappa _(alpha) > 0\kappa_\alpha>0,
Coercivity/compactness. Each budget is convex and l.s.c.; sublevel sets are equi-coercive after gauge-fixing (Ch. 4). Finite-block estimates lift by monotone convergence.
3.2 — Exact normalization via a unitary-path variational principle
We correct the stationarity→dynamics bridge by pricing the actual orbit speed|[H,A]|_(HS)|[H,A]|_{\rm HS} (not the Ad-invariant superoperator norm |delta _(H)|_(der)|\delta_H|_{\rm der}, whose directional derivative vanishes along conjugations).
3.2.A — Setup
Unitary paths and kinematics. Let U_(t)U_t be a strongly continuous unitary path with U_(0)=1U_0=\mathbf{1} and generator H_(t)=H_(t)^(†)H_t=H_t^\dagger on D\mathcal{D}:
Variations. Admissible variations are deltaU_(t)=-iK_(t)U_(t)\delta U_t=-\,iK_t U_t with K_(t)^(†)=K_(t)K_t^\dagger=K_t and K inC_(c)^(1)((0,T))K\in C_c^1((0,T)); then
Predictive score and gradient (HS metric). Let S:A_(loc)rarrR\mathcal{S}:\mathcal{A}_{\rm loc}\to\mathbb{R} be Gateaux-differentiable along commutators in the same normalized HS geometry:
Here lambda_(th) > 0\lambda_{\rm th}>0 is the throughput multiplier; using the same HS geometry will give ℏ=lambda_(th)^(-1)\hbar=\lambda_{\rm th}^{-1}.
Orbit Laplacian. For Hermitian AA on a finite block,
In an eigenbasis A=sum _(k)alpha _(k)P_(k)A=\sum_k\alpha_k P_k,
(A_(A)X)_(ij)=(alpha _(i)-alpha _(j))^(2)X_(ij),quad ker A_(A)={X:[A,X]=0},quadA_(A)^(+)" is the Moore–Penrose pseudoinverse."(\mathcal{A}_A X)_{ij}=(\alpha_i-\alpha_j)^2 X_{ij},\quad \ker \mathcal{A}_A=\{X:[A,X]=0\},\quad \mathcal{A}_A^+\text{ is the Moore–Penrose pseudoinverse.}
No-go for superoperator-norm pricing.H|->|delta _(H)|_(der;Lambda)H\mapsto |\delta_H|_{{\rm der};\Lambda} is Ad-invariant, hence the directional derivative of (1)/(2)|delta _(H)|_(der;Lambda)^(2)\frac{1}{2}|\delta_H|_{{\rm der};\Lambda}^2 along i[K,H]i[K,H] vanishes blockwise and in the inductive limit. Stationarity based on (:delta _(H),delta_(i[K,H]):)_(der)\langle\delta_H,\delta_{i[K,H]}\rangle_{\rm der} cannot produce a nontrivial G=lambda_(th)HG=\lambda_{\rm th}H.
Statement.
Under the setup above, let U_(*)^(**)U_\cdot^* be an interior maximizer of J\mathfrak{J} on [0,T][0,T] with H_(t)^(**)=H^(**)(A_(t))H_t^*=H^*(A_t) uniformly form-bounded on D\mathcal{D} and ||[H_(t)^(**),A_(t)]||_(HS)inL^(2)\|[H_t^*,A_t]\|_{\rm HS}\in L^2.
Then for all A inA_(loc)A\in\mathcal{A}_{\rm loc},
(i) Stationarity in HH.
Using (:[X,A],[Y,A]:)_(HS)=(:X,A_(A)Y:)_(HS)\langle [X, A], [Y, A] \rangle_{\rm HS} = \langle X, \mathcal{A}_A Y \rangle_{\rm HS} and ad_(A)^(**)=ad_(A)\mathrm{ad}_A^* = \mathrm{ad}_A,
0=del _(H)H=-lambda_(th)A_(A)H+iad_(A)P quad=>quadA_(A)H=(i)/(lambda_(th))ad_(A)P.0 = \partial_H \mathscr{H} = -\lambda_{\rm th} \mathcal{A}_A H + i \mathrm{ad}_A P \quad \Rightarrow \quad \boxed{\mathcal{A}_A H = \frac{i}{\lambda_{\rm th}} \mathrm{ad}_A P}.
(ii) Costate equation.
For orbit-tangent directions delta A=i[K,A]\delta A = i [K, A],
del _(A)H[i[K,A]]=(:G(A),K:)_(HS)-lambda_(th)(:[H,A],[H,i[K,A]]:)_(HS)+(:P,i[H,i[K,A]]:)_(HS).\partial_A \mathscr{H} [i [K, A]] = \langle G(A), K \rangle_{\rm HS} - \lambda_{\rm th} \langle [H, A], [H, i [K, A]] \rangle_{\rm HS} + \langle P, i [H, i [K, A]] \rangle_{\rm HS}.
Using HS-adjointness of ad_(H)\mathrm{ad}_H and Jacobi, this equals (:-,ad_(A)G(A)-iad_(A)ad_(H)P,K:)_(HS)\langle -, \mathrm{ad}_A G(A) - i \mathrm{ad}_A \mathrm{ad}_H P, K \rangle_{\rm HS}.
Hence
Projecting onto Ranad_(A)\mathrm{Ran} \mathrm{ad}_A (HS-orthogonal to the commutant) gives
ad_(A)P=-i[A,G(A)].\boxed{\mathrm{ad}_A P = -i [A, G(A)]}.
(iii) Eliminate PP.
Substitute into (i):
A_(A)H=(i)/(lambda_(th))ad_(A)P=(1)/(lambda_(th))(-i[A,G(A)]).\mathcal{A}_A H = \frac{i}{\lambda_{\rm th}} \mathrm{ad}_A P = \frac{1}{\lambda_{\rm th}} \big( -i [A, G(A)] \big).
The minimal-norm solution (orthogonal to ker A_(A)\ker \mathcal{A}_A) is H^(**)(A)=A_(A)^(+)((-i)/(lambda_(th))[A,G(A)])H^*(A) = \mathcal{A}_A^{+} \big( \tfrac{-i}{\lambda_{\rm th}} [A, G(A)] \big).
The state equation A^(˙)=i[H,A]\dot{A} = i [H, A] then yields the Heisenberg law with ℏ=lambda_(th)^(-1)\hbar = \lambda_{\rm th}^{-1}. ◻\square
Auxiliary calculus (explicit deltaJ\delta \mathfrak{J} on unitary paths).
On a fixed block (finite dd), let K inC_(c)^(1)((0,T))K \in C_c^1((0,T)) and use the identities, valid for Hermitian A,H,KA, H, K:
The last identity follows from B_(t):=ad_(A_(t))B_t := \mathrm{ad}_{A_t}, B^(˙)_(t)=ad_(A^(˙)_(t))=i[ad_(H_(t)),B_(t)]\dot{B}_t = \mathrm{ad}_{\dot{A}_t} = i [\mathrm{ad}_{H_t}, B_t] and (B_(t)^(2))^(˙)=i[ad_(H_(t)),B_(t)^(2)]\dot{(B_t^2)} = i [\mathrm{ad}_{H_t}, B_t^2], hence (d)/(dt)(B_(t)^(2)H_(t))=i[H_(t),B_(t)^(2)H_(t)]+B_(t)^(2)H^(˙)_(t)\frac{d}{dt} (B_t^2 H_t) = i [H_t, B_t^2 H_t] + B_t^2 \dot{H}_t.
Since KK is arbitrary in C_(c)^(1)C_c^1, stationarity of J\mathfrak{J}on unitary paths enforces A_(A)H^(˙)=-(1//lambda_(th))G(A)\mathcal{A}_A \dot{H} = -(1/\lambda_{\rm th}) G(A) on RanA_(A)\mathrm{Ran} \mathcal{A}_A.
This explicit computation shows why the earlier shortcut to (:G+lambda_(th)A_(A)H,K:)_(HS)\langle G + \lambda_{\rm th} \mathcal{A}_A H, K \rangle_{\rm HS} is invalid and justifies, for dynamics, the pointwise/Pontryagin route used in the proof above.
Normalization and units.
Because the same normalized HS geometry defines (i) the predictive gradient and (ii) the orbit-throughput quadratic, the physical Planck constant is fixed by
Calibration can be done operationally (e.g., two-level Fubini–Study speed) without ambiguity (§3.2.C).
3.2.C — Multiplier stability & metric uniqueness
Epi/Mosco stability across blocks.
With B_(orb;Lambda)(A,H):=(1)/(2)|[H,A]|_(HS;Lambda)^(2)\mathcal{B}_{{\rm orb};\Lambda}(A,H) := \tfrac{1}{2} |[H,A]|_{{\rm HS};\Lambda}^2 and B_(orb):=s u p _(Lambda)B_(orb;Lambda)\mathcal{B}_{\rm orb} := \sup_\Lambda \mathcal{B}_{{\rm orb};\Lambda}, the value functions
epi-converge (Mosco) to Phi\Phi defined with B_(orb)\mathcal{B}_{\rm orb}.
At points of differentiability, multipliers converge: lambda_(th,Lambda)rarrlambda_(th)\lambda_{{\rm th},\Lambda} \to \lambda_{\rm th}.
Hence ℏ=lambda_(th)^(-1)\hbar = \lambda_{\rm th}^{-1} is independent of the exhausting sequence.
Ad-invariant metric uniqueness (up to scale).
On each finite block, any Ad-invariant inner product on observables is a scalar multiple alpha\alpha of normalized HS (Schur).
Replacing (:*,*:)_(HS)\langle \cdot, \cdot \rangle_{\rm HS} by alpha(:*,*:)_(HS)\alpha \langle \cdot, \cdot \rangle_{\rm HS} rescales G|->alpha^(-1)GG \mapsto \alpha^{-1} G and |[H,A]|^(2)|->alpha|[H,A]|^(2)|[H,A]|^2 \mapsto \alpha |[H,A]|^2.
The product alphalambda_(th)^(-1)\alpha \lambda_{\rm th}^{-1} — i.e., ℏ\hbar — is invariant under this rescaling.
Thus ℏ\hbar is well-defined (after a single operational calibration).
3.3 — Hypotheses U (unbounded generators; GKSL on a core)
U1 (Core & closability). There exists a common invariant core D\mathcal{D} for HH and all L_(j)L_j; delta _(H)\delta_H is closable on A_(loc)\mathcal{A}_{\rm loc}.
U2 (Form bounds). There exists N >= 0N \ge 0 (number-type) and constants a < 1a < 1, b < oob < \infty with, for all psi inD\psi \in \mathcal{D},
||H psi||^(2)+sum _(j)||L_(j)psi||^(2) <= a||N psi||^(2)+b||psi||^(2).\|H \psi\|^2 + \sum_j \|L_j \psi\|^2 \le a \|N \psi\|^2 + b \|\psi\|^2.
U3 (Quasi-locality). Interactions have finite range and uniformly bounded overlap across Lambda\Lambda.
U4 (Semigroup). The GKSL closure generates a unique strongly continuous CPTP semigroup with Lieb–Robinson-type bounds.
U5 (Lyapunov drift). There exist c_(0),c_(1) > 0c_0, c_1 > 0 with L^(**)(N) <= c_(0)-c_(1)N\mathcal{L}^*(N) \le c_0 - c_1 N on D\mathcal{D}.
Drift ⇒ leakage finiteness.
For W=(1+N)^(-s)W = (1+N)^{-s} with s > 1//2s > 1/2,
sum _(j)omega(L_(j)^(†)WL_(j)) < ooquad=>quadB_(leak)({L_(j)},W) < oo,\sum_j \omega(L_j^\dagger W L_j) < \infty \quad \Rightarrow \quad \mathcal{B}_{\rm leak}(\{L_j\}, W) < \infty,
by Cauchy–Schwarz in the omega\omega-HS norm and U5.
3.4 — Zero-leakage ⇒ unitary GKSL
Let L=i[H,*]+sum _(j)L_(j)^(†)(*)L_(j)-(1)/(2){L_(j)^(†)L_(j),*}\mathcal{L} = i [H, \cdot] + \sum_j L_j^\dagger (\cdot) L_j - \tfrac{1}{2} \{L_j^\dagger L_j, \cdot\} satisfy U1–U4, and let W>-0W \succ 0.
Proposition 3.4.1. If B_(leak)({L_(j)},W)=sum _(j)omega(L_(j)^(†)WL_(j))=0\mathcal{B}_{\rm leak}(\{L_j\}, W) = \sum_j \omega(L_j^\dagger W L_j) = 0, then L_(j)=0L_j = 0 for all jj and the semigroup is unitary with generator HH on the core.
Proof. Each term omega(L_(j)^(†)WL_(j))=|W^(1//2)L_(j)|_(2,omega)^(2) >= 0\omega(L_j^\dagger W L_j) = |W^{1/2} L_j|_{2,\omega}^2 \ge 0. If the sum vanishes, W^(1//2)L_(j)=0W^{1/2} L_j = 0; since W>-0W \succ 0, L_(j)=0L_j = 0. ◻\square
Thus, in the zero-leakage limit, the fast sector is purely unitary, governed by HH from §3.2.
3.5 — Pointer alignment (unitary-orbit minimizer)
Fix W>-0W \succ 0. Consider a noise block with fixed singular values (fixed “strength spectrum”). For CC in this block define
L(C):=Tr(W^(1//2)C^(†)CW^(1//2)).\mathcal{L}(C) := \operatorname{Tr} \big( W^{1/2} C^\dagger C \, W^{1/2} \big).
Theorem 3.5.1 (unitary-orbit rearrangement). Over the orbit {UCV:U,V" unitary"}\{U C V : U, V \text{ unitary}\},
L(UCV)" is minimized iff "[C^(†)C,W]=0.\boxed{\mathcal{L}(U C V) \text{ is minimized iff } [C^\dagger C, W] = 0.}
Equivalently, minimizers co-diagonalizeC^(†)CC^\dagger C and WW, selecting the pointer basis (degeneracies handled blockwise).
Proof.L(UCV)=Tr(W^(1//2)V^(†)C^(†)CVW^(1//2))\mathcal{L}(U C V) = \operatorname{Tr}(W^{1/2} V^\dagger C^\dagger C V W^{1/2}). Let W=sum _(k)w_(k)Q_(k)W = \sum_k w_k Q_k with w_(1) >= cdots >= w_(d) > 0w_1 \ge \cdots \ge w_d > 0. By von Neumann/Schur–Horn, with fixed eigenvalues {sigma_(ℓ)}\{\sigma_\ell\} of C^(†)CC^\dagger C, the minimum of Tr(W,V^(†)C^(†)CV)\operatorname{Tr}(W, V^\dagger C^\dagger C V) is attained when VV aligns eigenvectors so that C^(†)CC^\dagger C is diagonal in the eigenbasis of WW. Equality enforces [C^(†)C,W]=0[C^\dagger C, W] = 0. ◻\square
3.6 — Summary
Pricing the orbit speed $|[H,A]|_{\rm HS}$ (rather than the Ad-invariant superoperator norm) and matching the same normalized HS geometry for gradient and quadratic yields
Here $\mathcal A_A=[A,[A,\cdot]]\ge0$ and $\mathcal A_A^+$ projects off the commutant.
Stability. Epi/Mosco limits across blocks preserve $\lambda_{\rm th}$; Ad-invariant metric rescalings cancel in $\hbar$. A single operational calibration fixes $\hbar$.
Fast sector. Zero leakage forces GKSL to reduce to a unitary group. For nonzero leakage, pointer alignment minimizes $\operatorname{Tr}(W{1/2}C\dagger C W^{1/2})$ by co-diagonalizing $C^\dagger C$ with $W$.
This completes the fast-sector normalization and resolves the stationarity/dynamics gap with explicit variational calculus and a pointwise/Pontryagin control derivation in a C*-compatible, mathematically airtight manner.
Scope. We state a clean cone‑preserving topology after gauge‑fixing and prove equi‑coercivity and Γ‑compactness. This slots into the EH Γ‑limit derivation and enforces single‑cone microhyperbolicity.
4.1 Γ‑compactness in a cone‑preserving class (Theorem 4.1′)
Setup. Bounded‑geometry manifold: uniform injectivity radius and bounded curvature (all derivatives). Fix de Donder gauge inside the single‑cone class; use harmonic coordinates on charts.
Cone-preserving topology.g inH_(loc)^(2)g \in H^2_{\rm loc} and for reference metrics g_(**),g^(**)g_*, g^* on compacts, g_(**) <= g <= g^(**)quad"a.e. in harmonic coordinates (cone inclusion)."g_* \le g \le g^* \quad \text{a.e. in harmonic coordinates (cone inclusion).}
Uniform symbol bounds. There exist 0 < lambda <= Lambda0<\lambda\le\Lambda such that for all admissible gg and xi\xi, lambda|xi|^(4) <= sigma(A_(epsi)(g))[xi] <= Lambda|xi|^(4)qquad(parameter‑ellipticity on compacts).\lambda\,|\xi|^4\ \le\ \sigma(\mathbb A_\varepsilon(g))[\xi]\ \le\ \Lambda\,|\xi|^4\qquad\text{(parameter‑ellipticity on compacts).}‑
Gårding inequality (uniform). There exist c_(1),c_(2),c_(3) > 0c_1,c_2,c_3>0, independent of epsi\varepsilon and gg, with F_(epsi)(g) >= c_(1)||grad^(2)g||_(L^(2))^(2)-c_(2)||g||_(H^(1))^(2)-c_(3).\boxed{\ \mathcal F_\varepsilon(g)\ \ge\ c_1\|\nabla^2 g\|_{L^2}^2\ -\ c_2\|g\|_{H^1}^2\ -\ c_3\ }.
Boundary/at‑infinity control. If MM is noncompact, either (i) impose uniform equivalence to a reference g_( oo)g_\infty outside a compact set, or (ii) include in V_( epsi)V_\varepsilon a cutoff penalizing deviations at infinity to control ||g||_(H^(1))\|g\|_{H^1}.
Theorem 4.1′ (equi‑coercivity and Γ‑limit)
The family {F_(epsi)}\{\mathcal F_\varepsilon\} is equi‑coercive in weak H^(2)H^2; any weak H^(2)H^2 cluster point of minimizers is a minimizer of F_(0)\mathcal F_0 (Γ‑liminf and recovery hold).
Proof sketch. Gauge‑fixing removes diffeomorphism degeneracy; bounded geometry yields Rellich compactness in weak H^(2)H^2; single‑cone stability controls the principal symbol. Γ‑compactness follows from equi‑coercivity and lower semicontinuity; recoveries are built by mollification in harmonic charts and partition‑of‑unity gluing.
Scope. We prove the coupled slow–fast laws at joint stationarity of the coherence program. The argument rests on:
a directional envelope theorem for worst-case pokes with direction-selecting minimizers (no illicit inf–variation swap);
effective sources T^(eff),J^(eff)T^{\rm eff},J^{\rm eff} defined first in the Clarke framework (existence + conservation without linearity), then upgraded to unique tensors under an explicit Gateaux/Clarke condition;
first-variation convergence for the slow Γ-limit via localized Mosco/Attouch;
a fast-sector control law that prices the derivation[H,*][H,\cdot] (the actual throughput), uses the Moore–Penrose pseudoinverse of the orbit Laplacian A_(A)=ad_(A)^(**)ad_(A)\mathcal A_A=\mathrm{ad}_A^{\,*}\mathrm{ad}_A, and yields the Heisenberg law with a consistent normalization ℏ=lambda_(th)^(-1)\hbar=\lambda_{\rm th}^{-1}.
5.1 — Admissible variables and objective
Slow variables. Lorentzian metrics gg of bounded geometry in the single-cone class (weak H_(loc)^(2)H^2_{\rm loc}, de Donder gauge on compacta) and compact-group connections AA with curvature F inL_(loc)^(2)F\in L^2_{\rm loc}.
Fast variables. A GKSL generator on the quasi-local C^(**)C^*-algebra (Ch. 3):
E1 (attainment/compactness). For each (g,A)(g,A), Phi|->CL(g,A;Phi)\Phi\mapsto \mathrm{CL}(g,A;\Phi) is l.s.c. and inf-compact on bar(P)\overline{\mathcal P}; hence M(g,A):=Argmin_(Phi in bar(P))CL(g,A;Phi)\mathsf M(g,A):=\operatorname{Argmin}_{\Phi\in\overline{\mathcal P}}\mathrm{CL}(g,A;\Phi)
is non-empty and compact.
E2 (Carathéodory).(g,A,Phi)|->CL(g,A;Phi)(g,A,\Phi)\mapsto \mathrm{CL}(g,A;\Phi) is continuous in (g,A)(g,A) for fixed Phi\Phi, and l.s.c. in Phi\Phi for fixed (g,A)(g,A).
E3 (directional differentiability). For each Phi\Phi and cone-admissible variation delta=(delta g,delta A)\delta=(\delta g,\delta A) with compact support,
exists and is positively homogeneous in delta\delta.
E4 (equi-differentiability on argmin graphs). There exist M > 0M>0 and a neighborhood U∋(g,A)U\ni(g,A) such that |DCL(g^('),A^(');Phi;delta)| <= M||delta|||D\,\mathrm{CL}(g',A';\Phi;\delta)|\le M\|\delta\|
for all (g^('),A^('))in U(g',A')\in U, Phi inM(g^('),A^('))\Phi\in\mathsf M(g',A'), and admissible delta\delta.
E5 (direction-wise selection). For each (g,A)(g,A) and direction delta\delta, choose
Corollary 5.2b (Gateaux differentiability — two routes)
At (g,A)(g,A), VV is Gateaux differentiable (two-sided, linear in delta\delta) if either:
(Unique active). A strictly convex, cone-local tie-breaker (vanishing in the Γ-limit) makes M(g,A)\mathsf M(g,A) a singleton with locally Lipschitz dependence; then DV(g,A;delta)=DCL(g,A;Phi^(**)(g,A);delta)D V(g,A;\delta)=D\,\mathrm{CL}(g,A;\Phi^*(g,A);\delta).
(Clarke-regular active set). M(g,A)\mathsf M(g,A) is compact and DCL(g,A;Phi;delta)D\,\mathrm{CL}(g,A;\Phi;\delta) is constant over Phi inM(g,A)\Phi\in\mathsf M(g,A) for each delta\delta. Then right/left derivatives coincide and are linear.
5.3 — Effective stress tensor and current (Clarke framework → upgrade)
We first ensure existence and conservation of sources without assuming linearity (Clarke framework), then upgrade to unique tensors under Cor. 5.2b.
Assumption L0 (local Lipschitz of the envelope)
Under E1–E4, the value function V(g,A)=min_(Phi inM(g,A))CL(g,A;Phi)V(g,A)=\min_{\Phi\in\mathsf M(g,A)}\mathrm{CL}(g,A;\Phi) is locally Lipschitz on the cone-class (Danskin–Rockafellar with the uniform bound in E4).
Localization for Clarke calculus. All Clarke sub/super-differential objects are taken on bounded subdomains Omega⋐M\Omega\Subset M with compactly supported variations; conclusions pass to MM by exhaustion Omega _(n)uarr M\Omega_n\uparrow M and compatibility of the cone-class cutoffs. This keeps the Banach-space hypotheses crisp and avoids overreach beyond bounded domains.
Clarke subdifferentials. Let del_(g)^(C)V(g,A)subH_(loc)^(-2)\partial_g^{\mathrm C} V(g,A)\subset H^{-2}_{\rm loc} and del_(A)^(C)V(g,A)subH_(loc)^(-1)\partial_A^{\mathrm C} V(g,A)\subset H^{-1}_{\rm loc} denote the Clarke subdifferentials; V_(g)^(@)(g,A;delta g)V_g^\circ(g,A;\delta g), V_(A)^(@)(g,A;delta A)V_A^\circ(g,A;\delta A) are the Clarke directional derivatives.
5.3.1 Subgradient sources (always well-posed) and conservation
Take any T^(eff)indel_(g)^(C)V(g,A)T^{\rm eff}\in\partial_g^{\mathrm C}V(g,A), J^(eff)indel_(A)^(C)V(g,A)J^{\rm eff}\in\partial_A^{\mathrm C}V(g,A) so that
Integrating by parts within the cone-class yields the weak conservation laws
grad ^(mu)T_(mu nu)^(eff)=0,qquadD^( mu)J_( mu)^(eff)=0\nabla^\mu T^{\rm eff}_{\mu\nu}=0,\qquad D^\mu J^{\rm eff}_\mu=0
for every Clarke-selectionT^(eff),J^(eff)T^{\rm eff},J^{\rm eff}.
5.3.2 Upgrade to single tensors (Distributional representation (Riesz on bounded domains); symmetry)
Assumption L (linearity at (g,A)(g,A)). One of Cor. 5.2b’s conditions holds at (g,A)(g,A). Then VV is Gateaux differentiable in g,Ag,A; the Clarke subdifferentials are singletons and coincide with the Fréchet derivatives.
Corollary. Under Assumption L there exist unique distributions T^(eff)inH_(loc)^(-2)T^{\rm eff}\in H^{-2}_{\rm loc}, J^(eff)inH_(loc)^(-1)J^{\rm eff}\in H^{-1}_{\rm loc} such that on each bounded domain Omega⋐M\Omega\Subset M
with T_(mu nu)^(eff)=T_(nu mu)^(eff)T^{\rm eff}_{\mu\nu}=T^{\rm eff}_{\nu\mu}; the representations are compatible under exhaustion and define T^(eff),J^(eff)T^{\rm eff},J^{\rm eff} globally on MM. Conservation from §5.3.1 persists.
5.4 — Slow-sector Euler–Lagrange: EH + YM (first-variation convergence)
Γ-convergence alone does not yield convergence of first variations. We adopt a localized Mosco/Attouch scheme.
Assumption M (localized Mosco/Attouch). On each bounded Omega⋐M\Omega\Subset M, F_(epsi)|_(Omega)\mathcal F_\varepsilon|_\Omega are equi-coercive, l.s.c., and admit integral representations with Carathéodory integrands f_( epsi)(x,*)f_\varepsilon(x,\cdot) obeying uniform growth/ellipticity and f_( epsi)rarrf_(0)f_\varepsilon\to f_0 in L^( oo)(Omega)L^\infty(\Omega). Then delF_(epsi)|_(Omega)rarr"graph"delF_(0)|_(Omega)\partial \mathcal F_\varepsilon|_\Omega \xrightarrow{\rm graph} \partial \mathcal F_0|_\Omega (Attouch). Exhaust Omega _(n)uarr M\Omega_n\uparrow M.
Lemma 5.4.1 (convergence of first variations). For any admissible (g,A)(g,A) and compactly supported cone-preserving (delta g,delta A)(\delta g,\delta A),
Theorem 5.4.2 (Einstein–YM with operational sources).
Let (g,A,H,{L_(j)})(g,A,H,\{L_j\}) be a cone-class KKT point of J\mathcal J and assume Assumption L at (g,A)(g,A). Then
with grad ^(mu)T_(mu nu)^(eff)=0\nabla^\mu T^{\rm eff}_{\mu\nu}=0 and D^( mu)J_( mu)^(eff)=0D^\mu J^{\rm eff}_\mu=0.
Proposition 5.4.3 (constants as multipliers = Γ-calibration).
Under Slater and strict convexity of F_(0)\mathcal F_0 along the cone-class, KKT multipliers (G^(-1),Lambda,g_(YM)^(-2))(G^{-1},\Lambda,g_{\rm YM}^{-2}) are unique. Calibrating F_(epsi)\mathcal F_\varepsilon on (i) Minkowski and (ii) constant-curvature backgrounds fixes the same constants; the Γ-calibrated constants coincide with the KKT duals.
5.5 — Fast-sector stationarity and microcausal hygiene
We price the derivation (the actual throughput) and solve the first-order condition on the unitary-orbit tangent using the Moore–Penrose pseudoinverse of the orbit Laplacian.
5.5.1 Unitary-orbit control with derivation-quadratic (Heisenberg law)
Work on a finite block Lambda\Lambda with normalized Hilbert–Schmidt (HS) inner product (:X,Y:)_(HS):=Tr_(Lambda)(X^(†)Y)//d_( Lambda)\langle X,Y\rangle_{\rm HS}:=\mathrm{Tr}_\Lambda(X^\dagger Y)/d_\Lambda. Let A_(t)=U_(t)^(†)A_(0)U_(t)A_t=U_t^\dagger A_0U_t, U^(˙)_(t)=-(i)/(ℏ)H_(t)U_(t)\dot U_t=-\tfrac{i}{\hbar}H_tU_t, so
Let P(A)\mathsf P(A) be the orbit-restricted predictive score; G(A):=grad_(HS)P(A)G(A):=\operatorname{grad}_{\rm HS}\mathsf P(A). Key pairing (explicit HS calculation). Work on a finite block Lambda\Lambda with normalized Hilbert–Schmidt inner product (:X,Y:)_(HS):=Tr(X^(†)Y)//d_( Lambda)\langle X,Y\rangle_{\rm HS}:=\mathrm{Tr}(X^\dagger Y)/d_\Lambda. For Hermitian A,HA,H,
{:[(:[A","X]","Y:)_(HS)=(1)/(d_( Lambda))Tr((AX-XA)^(†)Y)],[=(1)/(d_( Lambda))Tr(X^(†)AY-AX^(†)Y)],[=(1)/(d_( Lambda))Tr(X^(†)AY-X^(†)YA)quad("cyclicity")],[=(:X","[A","Y]:)_(HS).]:}\begin{aligned}
\langle [A,X],Y\rangle_{\rm HS}
&= \frac{1}{d_\Lambda}\mathrm{Tr}\big((AX-XA)^\dagger Y\big) \\
&= \frac{1}{d_\Lambda}\mathrm{Tr}\big(X^\dagger A Y - A X^\dagger Y\big) \\
&= \frac{1}{d_\Lambda}\mathrm{Tr}\big(X^\dagger A Y - X^\dagger Y A\big) \quad(\text{cyclicity})\\
&= \langle X,[A,Y]\rangle_{\rm HS}.
\end{aligned}
Thus the commutator superoperator ad_(A):X|->[A,X]\mathrm{ad}_A:X\mapsto[A,X] is HS-self-adjoint on Hermitian AA. Along a unitary orbit A_(t)=U_(t)^(†)A_(0)U_(t)A_t=U_t^\dagger A_0 U_t with U^(˙)_(t)=-(i)/(ℏ)H_(t)U_(t)\dot U_t=-\tfrac{i}{\hbar}H_tU_t,
Let P(A)\mathsf P(A) be the orbit-restricted predictive score and G(A):=grad_(HS)P(A)G(A):=\operatorname{grad}_{\rm HS}\mathsf P(A). For any Hermitian control HH,
and the corresponding generator is A^(˙)=(i)/(ℏ)[H^(**)(A),A]\dot A=\tfrac{i}{\hbar}[H^*(A),A]. Local Lipschitz and blockwise bounds propagate to the quasi-local algebra via the uniform HS-block controls of Ch. 3.
5.5.2 Leakage and GKSL
Allow Lindblad controls {L_(j)}\{L_j\} with leakage price lambda_(leak)sum _(j)||W^(1//2)L_(j)||_(HS)^(2)\lambda_{\rm leak}\sum_j\|W^{1/2}L_j\|_{\rm HS}^2 (App. A). The convex optimization over CP-tangent directions yields an optimal GKSL generator with leakage-penalized pointer alignment as in Ch. 3. Hypotheses U give well-posedness and Lieb–Robinson-type bounds on the cone-class.
5.5.3 Microcausal hygiene
The poke cone, orbit locality, and LR-type bounds ensure commutator growth remains inside the cone; the principal-symbol lemma (§4.1) forbids order flips without paying the W_(1)W_1 gap. All variational steps above remain legal within the cone-preserving class.
5.6 — Summary (coupled laws)
At joint KKT stationarity of J\mathcal J on the cone-preserving class:
with T^(eff),J^(eff)T^{\rm eff},J^{\rm eff}well-posed as Clarke subgradients (conserved for every selection).
Under Assumption L (unique minimizer or Clarke-regular active set with constant directional derivatives), they upgrade to single linear distributions with distributional (Riesz-on-bounded-domains) representation, T_(mu nu)^(eff)=T_(nu mu)^(eff)T^{\rm eff}_{\mu\nu}=T^{\rm eff}_{\nu\mu}, grad ^(mu)T_(mu nu)^(eff)=0\nabla^\mu T^{\rm eff}_{\mu\nu}=0, D^( mu)J_( mu)^(eff)=0D^\mu J^{\rm eff}_\mu=0.
With leakage, the optimal generator is GKSL with leakage-penalized pointer alignment; well-posed with LR-type bounds.
Constants.ℏ,G,Lambda,g_(YM)\hbar,\,G,\,\Lambda,\,g_{\rm YM} are the unique KKT multipliers and match the Γ-calibrated constants (Prop. 5.4.3).
All steps are now rigorous and consistent with the coherence-budget framework: directional envelope calculus with direction-selecting minimizers; Clarke-sound sources and conservation (upgraded to tensors when linearity holds); first-variation convergence for the slow Γ-limit; and a derivation-quadratic control law on the unitary orbit (with pseudoinverse and explicit normalization) producing the Heisenberg/GKSL fast dynamics.
Scope. With budgets and cone hygiene in place, we summarize the selection for the gauge scaffold and record the hypercharge‑uniqueness result under the standard binders (single Higgs doublet; anomaly cancellation; minimal chiral set). Proofs and linear‑system details mirror Appendix C.
6.1 Budget–symmetry selection (sketch)
Ad‑invariant quadratic forms on a compact simple Lie algebra are multiples of the Killing form; additivity across factors gives sum _(i)C_(2)(G_(i))\sum_i C_2(G_i). The unique Ad‑invariant count surcharge is sum _(i)dim G_(i)\sum_i\dim G_i. These, with Γ‑locality and mediator locality, generate the mediator part of the complexity budget B_(cx)B_{\rm cx}. Minimal coupling follows from the Γ‑limit of the connection scaffold.
Assumptions for hypercharge (conditional uniqueness).
Uniqueness here is conditional on the binder set: (B_Yuk) renormalizable Yukawas, (B_{\rm anom}) anomaly cancellation, and (B_{\rm min}) the minimal chiral content with a single Higgs doublet. Relaxing any binder re-opens branches; see Appendix C for alternatives and their costs.
Binders. (i) B_Yuk: only renormalizable Yukawas QHd^(c),Q tilde(H)u^(c),LHe^(c)QH d^c,\ Q\tilde H u^c,\ L H e^c. (ii) B_anom: cancel [SU(3)]^(2)U(1)_(Y),[SU(2)]^(2)U(1)_(Y),U(1)_(Y)^(3),"grav"^(2)-U(1)_(Y)[SU(3)]^2U(1)_Y,\ [SU(2)]^2U(1)_Y,\ U(1)_Y^3,\ \text{grav}^2\!\!\!-\!U(1)_Y anomalies. (iii) B_min: one‑generation minimal chiral set {Q,u^(c),d^(c),L,e^(c)}\{Q,u^c,d^c,L,e^c\}; one Higgs doublet.
Statement. Let (Y_(Q),Y_(u),Y_(d),Y_(L),Y_(e),Y_(H))(Y_Q,Y_u,Y_d,Y_L,Y_e,Y_H) be unknown hypercharges. Under B_Yuk + B_anom + B_min, the solution set is a one‑parameter line (overall normalization/orientation). Fixing the unit with the minimal‑charge binder (Q=T_(3)+YQ=T_3+Y and color‑singlet integrality) yields the Standard Model values (Y_(Q),Y_(u),Y_(d),Y_(L),Y_(e),Y_(H))=((1)/(6),-(2)/(3),(1)/(3),-(1)/(2),1,-(1)/(2)).\boxed{(Y_Q,Y_u,Y_d,Y_L,Y_e,Y_H)=\big(\tfrac16,-\tfrac23,\tfrac13,-\tfrac12,1,-\tfrac12\big)}.
Proof (linear system & rank). Yukawa invariance gives Y_(Q)+Y_(H)+Y_(d)=0Y_Q+Y_H+Y_d=0, Y_(Q)-Y_(H)+Y_(u)=0Y_Q-Y_H+Y_u=0, Y_(L)+Y_(H)+Y_(e)=0Y_L+Y_H+Y_e=0. Anomalies add 3Y_(Q)+Y_(L)=03Y_Q+Y_L=0 and 6Y_(Q)+3Y_(u)+3Y_(d)+2Y_(L)+Y_(e)=06Y_Q+3Y_u+3Y_d+2Y_L+Y_e=0. Solve: Y_(H)=-3Y_(Q)Y_H=-3Y_Q, Y_(L)=-3Y_(Q)Y_L=-3Y_Q, Y_(e)=+6Y_(Q)Y_e=+6Y_Q, Y_(u)=-4Y_(Q)Y_u=-4Y_Q, Y_(d)=+2Y_(Q)Y_d=+2Y_Q with Y_(Q)Y_Q free (rank 5/6). Minimal‑charge normalization via Q=T_(3)+YQ=T_3+Y and Q(nu)=0Q(\nu)=0 fixes Y_(Q)=1//6Y_Q=1/6. After normalization, strict convexity/tie‑breakers on B_(cx)B_{\rm cx} exclude co‑minima; the SM assignment is unique with a positive gap.
Scope. We work in a local Killing-horizon patch with surface gravity $\kappa>0$, the single-cone class and the C*-compatible budgets of Ch. 3, Γ-compact slow sector of Ch. 4, and the coupled-laws framework of Ch. 5. We:
(i) specify a near-horizon Unruh-diagonal GKSL model fixed by pointer alignment;
(ii) take mutual information $I(A:\bar A)$ as the primary entropy functional (with $E_R\le I$), prove quasi-factorization with explicit LR-dependent constants;
(iii) give a per-tile information–budget bound controlled by the leakage budget; define a pointer cutoff $\ell_*$ by KKT thresholding; and prove an area-law upper bound with a calibrated constant;
(iv) prove a sharp flux-suppression inequality as a linear program in the rates, in full generality and with an exact Hawking-weight corollary;
(v) record microcausality and an identifiability lemma to determine Hawking rates from pointer-basis two-point data.
Proof details and constant tracking are in Appendix G (area law) and Appendix H (flux suppression). KPI & refuter. The near-horizon prediction is a universal amplitude suppression of the outgoing flux by a budget-controlled factor at fixed Hawking temperature. Any observed temperature shift at leading order would falsify the budget-consistent GKSL picture; amplitude-only suppression with the predicted frequency shaping supports it.
7.1 Near-horizon setup, pointer alignment, and quasi-factorization
7.1.1 Geometry and Unruh modes
Fix a stationary Killing horizon with surface gravity $\kappa>0$. On a bounded chart of the horizon neighborhood we use Rindler coordinates to the accuracy guaranteed by Ch. 4 bounded-geometry hypotheses. Let
Lemma 7.1 (Budget-monotonic pointer alignment).
Let L_(T)\mathcal{L}_T be any GKSL on TT with the same singular values of the noise block as above. Let Delta\Delta be the pinching (full dephasing) in the Unruh basis. Then:
(Budget) sum _(j)omega(L_(j)^(†)WL_(j)) >= sum _(j)omega((DeltaL_(j))^(†)W(DeltaL_(j)))\sum_j \omega(L_j^\dagger W L_j) \ge \sum_j \omega \big( (\Delta L_j)^\dagger W (\Delta L_j) \big) (Hilbert-Schmidt pinching contraction).
(Information) For any state rho\rho, I((idoxLambda _(t))(rho): bar(T))I\big( (\mathrm{id} \otimes \Lambda_t)(\rho) : \bar{T} \big)does not increase if one replaces Lambda _(t):=e^(tL_(T))\Lambda_t := e^{t \mathcal{L}_T} by Delta@Lambda _(t)@Delta\Delta \circ \Lambda_t \circ \Delta (monotonicity of II under local CPTP maps and commutation of Delta\Delta with the Unruh-diagonal semigroup).
Hence, among channels with fixed spectral data, Unruh-diagonal noise minimizes leakage cost and maximizesI(T: bar(T))I(T : \overline{T})only within that spectral class. We therefore restrict to the diagonal form without loss for upper bounds.
Proof. (1) The map X|->W^(1//2)XX \mapsto W^{1/2} X followed by conditional expectation Delta\Delta is a contraction in the HS norm; sum over jj. (2) Mutual information is monotone under local CPTP maps; Delta\Delta is local on TT and leaves the Unruh-diagonal dynamics invariant; compose. ◻\square
(no $\tfrac12$ factor in general). All subsequent bounds are proved for $I$ and then immediately transfer to $E_R$.
7.1.4 Quasi-factorization with LR constants
Tile a region AA by disjoint squares {T_(j)}_(j in J(A))\{T_j\}_{j\in J(A)} of side ℓ\ell, and include a boundary collar of width O(ℓ)O(\ell), yielding O(|del A|)O(|\partial A|) collar tiles. The GKSL semigroup obeys cone-limited LR bounds (Ch. 3 U4; Appendix F.2) and admits an Unruh thermal log-Sobolev (or spectral-gap) constant on each tile (Appendix G.1).
Theorem 7.0 (Quasi-factorization of mutual information).
There exist C_(LR),xi,v_(LR) < ooC_{\rm LR}, \xi, v_{\rm LR} < \infty, depending only on the LR/mixing data and the local mode density, such that for all ℓ >= ℓ_(0)\ell \ge \ell_0 and all horizon-patch regions AA,
In particular, choosing ℓ >= c,xi log(1+|del A|)\ell \ge c , \xi \log(1 + |\partial A|) absorbs the remainder into the boundary term: I(A: bar(A)) <= sum _(j)I(T_(j): bar(T_(j)))+O(|del A|)I(A : \bar{A}) \le \sum_j I(T_j : \overline{T_j}) + O(|\partial A|).
Proof (outline; Appendix G.2). Chain the mutual information by SSA: I(A: bar(A))=sum _(j)I(T_(j): bar(A),|T_( < j))I(A : \bar{A}) = \sum_j I \big( T_j : \bar{A} , \big| T_{<j} \big) and bound each conditional term by an LR-decaying influence from bar(T_(j))\overline{T_j} outside a collar of width ∼ℓ\sim \ell. Mix to Unruh steady on the collar using the tile log-Sobolev gap; constants track to C_(LR),xiC_{\rm LR}, \xi. ◻\square
7.2 Area-law upper bound with calibrated constant
We now reduce the area bound to a per-tile estimate and calibrate the constant.
7.2.1 Per-tile information vs. leakage budget
For a tile TT, write Gamma _(k):=gamma _(k)+ tilde(gamma)_(k)\Gamma_k := \gamma_k + \tilde{\gamma}_k and define
Proposition 7.1 (Linear upper bound, sharp coefficient).
There exists a finite constant
chi _(T):=s u p_(k inK_(T))(del^(+)I_(k))/(delGamma _(k))(1)/(w_(k)),\chi_T := \sup_{k \in \mathcal{K}_T} \ \frac{\partial^+ I_k}{\partial \Gamma_k} \, \frac{1}{w_k} \, ,
(where I_(k)I_k is the single-mode contribution under Unruh-diagonal dynamics and del^(+)\partial^+ is the right derivative at the realized Gamma _(k)\Gamma_k) such that
Moreover chi _(T)\chi_T depends only on the Unruh temperature, the LR/mixing constants, and the local mode density; it is uniform across tiles of the same side ℓ\ell.
Proof (Appendix G.3). For Unruh-diagonal GKSL, I(T: bar(T))=sum _(k)I_(k)(Gamma _(k))I(T : \overline{T}) = \sum_k I_k(\Gamma_k). Each I_(k)I_k is concave, increasing in Gamma _(k)\Gamma_k and differentiable a.e. (data processing + semigroup contractivity in the normalized-HS metric used for budgets, Appendix E.1). Lagrange’s inequality then gives I(T: bar(T)) <= sum _(k)(del^(+)I_(k)//delGamma _(k))Gamma _(k) <= chi _(T)sum _(k)w_(k)Gamma _(k).I(T : \overline{T}) \le \sum_k (\partial^+ I_k / \partial \Gamma_k) \Gamma_k \le \chi_T \sum_k w_k \Gamma_k.
Uniformity follows from bounded geometry and the common Unruh temperature. ◻\square
Remark. We carry (7.2.1) as an upper bound. Under the Hawking-weight calibration of §7.3.2 the coefficient becomes tile-independent (chi _(T)-=chi_(**)\chi_T \equiv \chi_*) and the bound is tight (equality along the Hawking ray).
7.2.2 KKT thresholding and the pointer cutoff $\ell_*$
We optimize per-tile I(T: bar(T))I(T : \overline{T}) subject to the leakage allowance tau_(leak)(T)\tau_{\rm leak}(T). The KKT conditions with multiplier lambda_(leak) > 0\lambda_{\rm leak} > 0 give the threshold rule
Because w_(k)w_k grows monotonically with transverse momentum (pointer/energy scaling) and delI_(k)//delGamma _(k)\partial I_k / \partial \Gamma_k is bounded and decreases with |k||k| at fixed beta _(U)\beta_U, the optimizer activates modes only for |k|≲ℓ_(**)^(-1)|k| \lesssim \ell_*^{-1} with a sharp cutoff at
(Details in Appendix G.4.) For tiles with side ℓ=Theta(ℓ_(**))\ell = \Theta(\ell_*), the number of active transverse modes is ≃|T|//ℓ_(**)^(d-2)\simeq |T| / \ell_*^{d-2} up to boundary/collar corrections.
7.2.3 Area-law theorem
Theorem 7.1 (Area-law upper bound with boundary term).
For tiles of side ℓ=Theta(ℓ_(**))\ell = \Theta(\ell_*),
Here chi_(**):=s u p _(T)chi _(T)\chi_* := \sup_T \chi_T is finite and depends only on the Unruh temperature, LR/mixing constants and the local weight profile WW (Appendix G.5). The O(|del A|)O(|\partial A|) constant depends only on LR/geometry data.
Proof. Combine Theorem 7.0 with Proposition 7.1 and the thresholding definition of ℓ_(**)\ell_*; count active modes per interior tile and absorb LR remainders into the collar. ◻\square
7.2.4 Calibration to Einstein–Hilbert and the 1//41/4 coefficient
Proposition 7.2 (EH calibration =>\Rightarrowchi_(**)=(1)/(4)\chi_* = \tfrac{1}{4} in Planck units).
Under the Γ-limit normalization of Appendix D (Ch. 4/D.5), the slow action equals Einstein–Hilbert and the Unruh temperature is fixed by kappa\kappa. Matching the tile-wise Clausius relationdelta Q=T_(U)delta S\delta Q = T_U \delta S with the leakage work priced by WW at the pointer cutoff yields
Proof (Appendix G.6). The per-tile leakage expenditure that realizes the Unruh KMS structure at the cutoff equals the heatdelta Q\delta Q through the stretched horizon; with T_(U)T_U fixed, the delta S\delta S density matches EH’s area-entropy density, fixing chi_(**)=1//4\chi_* = 1/4. ◻\square
7.3 Hawking-flux suppression: sharp LP bound and calibrated equality
7.3.1 General linear-program bound (no profile assumptions)
Let c_(k)c_k denote the outgoing flux weight per mode (number or energy),
Proof. Linear objective over a simplex: maximize sumc_(k)Gamma _(k)\sum c_k \Gamma_k s.t. sumw_(k)Gamma _(k) <= tau\sum w_k \Gamma_k \le \tau and Gamma _(k) >= 0\Gamma_k \ge 0. KKT gives support on argmax of c_(k)//w_(k)c_k / w_k. ◻\square
Microcausality. The LR bounds (Appendix F.2) imply ||[Phi _(t)(O_(X)),O_(Y)]|| <= Ce^(-(d(X,Y)-v_(LR)t)//xi)\| [\Phi_t(O_X), O_Y] \| \le C e^{-(d(X,Y) - v_{\rm LR} t)/\xi}; hence no superluminal contribution to F_(out)\mathcal{F}_{\rm out} is admissible.
7.3.2 Hawking-weight calibration and exact multiplicative factor
i.e. choose the leakage weight to price exactly the flux contribution (number or energy). This choice is canonical operationally (Appendix J.1: weight normalization by KPI).
Let Gamma_(k)^(H)\Gamma_k^{\rm H} be the detailed-balance rates reproducing the semiclassical Hawking flux, and define the Hawking leakage cost
Proof. Under w_(k)^(H)propc_(k)w_k^{\rm H} \propto c_k, c_(k)//w_(k)^(H)c_k / w_k^{\rm H} is constant in kk; every Hawking-proportional allocation maximizes the LP; scale by feasibility. ◻\square
7.4 Identifiability, observables, and explicit $f$
7.4.1 Identifiability of Hawking rates in the pointer model
Lemma 7.3 (Pointer-basis identifiability).
In the Unruh-diagonal GKSL model, the two-point functions of the outside modes,
Hence the KMS ratiotilde(C)_(k)(t)//C_(k)(t)= bar(n)_(k)//(1+ bar(n)_(k))=e^(-beta _(U)omega _(k))\tilde{C}_k(t)/C_k(t) = \bar{n}_k / (1 + \bar{n}_k) = e^{-\beta_U \omega_k} fixes beta _(U)\beta_U, and the linewidth identifies Gamma _(k)\Gamma_k. In particular, the Hawking rates Gamma_(k)^(H)\Gamma_k^{\rm H} are uniquely determined from pointer-basis two-point data within this model class.
Proof. Solve the Heisenberg equations under the Unruh-diagonal Lindbladian; the amplitudes decay at rate (1)/(2)Gamma _(k)\tfrac{1}{2} \Gamma_k while detailed balance fixes the ratio. ◻\square
Tomography in the pointer basis (Appendix J.1) gives Gamma_(k)^(H)\Gamma_k^{\rm H} from linewidths and beta _(U)\beta_U from KMS.
Compute the Hawking costtau_(H)(W)=sumw_(k)Gamma_(k)^(H)\tau_{\rm H}(W) = \sum w_k \Gamma_k^{\rm H} for the chosen weight WW (or set W=W_(H)W = W_{\rm H} to make it canonical).
which solves to ℓ=Theta(ℓ_(**))\ell = \Theta(\ell_*) defined in §7.2.2. With this choice, the quasi-factorization remainder is absorbed into O(|del A|)O(|\partial A|) and the area coefficient is calibrated by Proposition 7.2.
7.5 Microcausality (guard) and hygiene
The single-cone class (Ch. 4) and U-assumptions (Ch. 3) yield LR bounds (Appendix F.2):
Thus neither the area-law derivation (local tilings and collar mixing) nor the flux LP can exploit superluminal influences; all optimizers live inside the cone.
7.6 Summary
Pointer alignment is budget-monotonic and information-preserving in the sense of Lemma 7.1; hence we restrict to Unruh-diagonal GKSL.
A quasi-factorization theorem (Theorem 7.0) holds with explicit LR constants, reducing the bound to per-tile contributions plus an O(|del A|)O(|\partial A|) boundary term.
Per-tile information obeys a linear upper boundI(T: bar(T)) <= chi _(T)tau_(leak)(T)I(T : \overline{T}) \le \chi_T \, \tau_{\rm leak}(T) (Proposition 7.1), with a KKT threshold producing a pointer cutoffℓ_(**)\ell_* (§7.2.2).
Summing tiles gives an area law with explicit coefficient chi_(**)//ℓ_(**)^(d-2)\chi_* / \ell_*^{d-2} (Theorem 7.1). Under EH Γ-calibration, chi_(**)=1//4\chi_* = 1/4 (Proposition 7.2), yielding
Identifiability (Lemma 7.3) pins Gamma_(k)^(H)\Gamma_k^{\rm H} from pointer-basis two-point data; microcausality is enforced by LR bounds.
All constants and intermediate inequalities are tracked in Appendix G (area) and Appendix H (flux), ensuring the chapter’s statements are fully rigorous within the stated hypotheses.
Scope. Define a coherence‑preserving coarse‑graining and derive an RG flow on budgets and predictive envelopes. Fixed points correspond to scale‑invariant scaffolds; relevant directions match active budgets.
8.1 Coarse‑graining & monotonicity
Local coarse‑grainings C_(ℓ)\mathcal C_\ell at scale ℓ\ell respect the poke cone (causal, Γ‑local). Budgets obey monotone inequalities B_(∙)(C_(ℓ)A) <= c_(∙)(ℓ)B_(∙)(A),quad∙in{th,cx,leak}.B_\bullet(\mathcal C_\ell A)\ \le\ c_\bullet(\ell)\, B_\bullet(A),\quad \bullet\in\{\rm th,cx,leak\}.
8.2 Flow equations (envelope picture)
Scale‑ℓ\ell objective V_(ℓ)(A)=i n f_(Phi in bar(P))CL(C_(ℓ)A,Phi)-sum∙lambda_(∙)B_(∙)(C_(ℓ)A)\mathcal V_\ell(A)= \inf_{\Phi\in\overline{\mathcal P}}\mathrm{CL}(\mathcal C_\ell A,\Phi) - \sum_\bullet \lambda_\bullet B_\bullet(\mathcal C_\ell A). The coherence RG is del_(lnℓ)V_(ℓ)=DV_(ℓ)-sum∙(del_(lnℓ)ln c_(∙))*lambda_(∙)B_(∙)+(irrelevant corrections).\partial_{\ln\ell}\, \mathcal V_\ell\ =\ \mathcal D\,\mathcal V_\ell\ -\sum_\bullet (\partial_{\ln\ell}\ln c_\bullet)\cdot \lambda_\bullet B_\bullet\ +\ \text{(irrelevant corrections)}.
8.3 Fixed points and stability
A fixed point satisfies stationarity of V_(ℓ)\mathcal V_\ell up to rescaling; budgets transform covariantly. Linearization gives scaling exponents for (th,cx,leak) channels and binders. The RG‑envelope yields closure rules for multipliers by matching to observed invariants.
Scope. We (i) compute the hypersurface-deformation (ADM/Henneaux–Teitelboim) algebra inside the cone-preserving class, verifying closure without anomalies and persistence at the Γ-limit, and (ii) construct a cone-limited projective path measure with explicit finite-dimensional marginals, proving cylinder consistency, FKG/positive association under an attractive discretization, and microcausality via LR-type bounds at the measure level. All hypotheses and objects align with the budgets/topologies fixed earlier (bounded geometry, de Donder/harmonic gauge on charts, single-cone hygiene, Γ-locality). BRST/BV hygiene note (anomaly absence at the Γ-limit). Within the cone-preserving, gauge-fixed class and bounded-geometry hypotheses, the hypersurface-deformation algebra closes without central extensions (Appendix D/F bounds). The corresponding BRST charge is nilpotent, and a unitary BV/BV measure exists for the projective path construction; microcausality (LR-type) guards ensure cylinder consistency.
9.1 Constraint algebra closure (ADM/Henneaux–Teitelboim inside the cone class)
Phase space, constraints, and smearings
Let $\Sigma$ be a Cauchy slice of a globally hyperbolic spacetime with bounded geometry. The canonical variables are the Riemannian metric $q_{ab}$ on $\Sigma$ and its conjugate momentum $\pi{ab}=\sqrt{q},(K{ab}-K q^{ab})$ (indices raised/lowered with $q$). The (Poisson) symplectic form is
The scalar (Hamiltonian) and vector (momentum) constraints (pure gravity with $\Lambda$; minimal coupling to gauge/matter adds standard pieces without modifying the algebraic structure functions) are
with $D$ the Levi-Civita connection of $q$, $R$ its scalar curvature, and $\pi:=q_{ab}\pi^{ab}$. For smearings $N\in C_c^\infty(\Sigma)$ and $N^a\in\mathfrak X_c(\Sigma)$ (compact support or appropriate falloff), define
All fields are restricted to the single-cone class (principal symbol bounds) and we work in harmonic/de Donder gauge on charts when needed (for well-posedness and elliptic control of gauge).
Hypotheses for this block
(C1) Cone hygiene & bounded geometry. As in Ch. 4, uniform injectivity radius and curvature bounds; single-cone principal-symbol bounds hold.
(C2) Boundary/falloff. Either compact Sigma\Sigma without boundary or standard ADM falloffs guaranteeing boundary terms vanish in the bracket computations (or are absorbed in the ADM surface charges if present; here we take vanishing boundaries for brevity).
(C3) Γ-limit stability. The slow action is the Γ-limit of second-order local forms (Ch. 4/Appendix D), so all variational derivatives converge in the sense needed below.
The hypersurface-deformation algebra (HDA): statements
i.e., closure with structure functions q^(ab)q^{ab} and no central extensions.
We prove (9.1) within the cone-preserving class and verify persistence at the Γ-limit.
Therefore, for any functional FF, {F,D[ vec(N)]}=L_( vec(N))F\{F, D[\vec{N}]\} = \mathcal{L}_{\vec{N}} F. Apply to F=D[ vec(M)]F = D[\vec{M}]; since DD is a spatial vector density of weight one,
For V[N]V[N], use delta(sqrtqR)=sqrtq(G^(ab)deltaq_(ab)+D_(c)Theta ^(c))\delta(\sqrt{q} R) = \sqrt{q} (G^{ab} \delta q_{ab} + D_c \Theta^c) with G^(ab)G^{ab} the Einstein tensor of qq and Theta ^(c)\Theta^c a boundary term; thus
Insert in the Poisson bracket and integrate by parts. Divergences vanish by (C2). Curvature terms combine with derivatives of N,MN,M to yield the shift vector
beta ^(a)[q;N,M]:=q^(ab)(Ndel _(b)M-Mdel _(b)N).\beta^a[q; N, M] := q^{ab} (N \partial_b M - M \partial_b N).
A direct (standard) but lengthy cancellation gives
Matter contributions are covariant and assemble into the same form (the momentum constraint is the Noether charge for spatial diffeomorphisms), hence do not alter the structure functions. No central term appears: any putative c-number must be a boundary integral antisymmetric in (N,M)(N,M), which vanishes under (C2). ◻\square
Gauge-fixing, Dirac brackets, and anomaly exclusion
Let chi ^(mu)=0\chi^\mu = 0 be a (cone-preserving) gauge, e.g. harmonic/de Donder on charts. The set {H_(_|_),H_(a),chi ^(mu)}\{\mathcal{H}_\perp, \mathcal{H}_a, \chi^\mu\} is second-class with invertible bracket matrix on the single-cone domain. The Dirac bracket{*,*}_(D)\{\cdot, \cdot\}_D equals the Poisson bracket on gauge-invariant functionals. Since H[N],D[ vec(N)]H[N], D[\vec{N}] generate diffeomorphisms, their algebra (9.1) remains valid with {*,*}_(D)\{\cdot, \cdot\}_D when acting on gauge-invariant observables.
Absence of anomalies at the Γ-limit. The slow action is a Γ-limit of second-order local functionals with uniform symbol bounds (Ch. 4). Variational derivatives of the discretized constraints converge (in the Mosco sense) to those of EH+YM; the symplectic form is fixed. Hence the structure functions converge to q^(ab)q^{ab}, and the brackets converge to (9.1). A central extension would require either (i) a cone flip (excluded by the W_(1)W_1 gap; App. F.1) or (ii) higher-derivative remnants violating H1/H5; both are ruled out by our hypotheses.
{:(9.2)"The constraint algebra closes (no central terms) in the cone-preserving class and persists at the Γ-limit.":}\boxed{\text{The constraint algebra closes (no central terms) in the cone-preserving class and persists at the Γ-limit.}}
\tag{9.2}
We build a cone-limited probability measure on histories of the slow fields (metric gg and gauge fields AA) compatible with the budgets and microcausality.
The construction is by projective limits of cylinder measures with explicit finite-dimensional marginals.
Indexing of cylinders and state spaces
Let P\mathscr{P} be the directed set of finite space–time partitionsPP: harmonic-chart coverings of compact slabs K xx[t_(0),t_(1)]K \times [t_0, t_1] with mesh (ℓ,Delta t)(\ell, \Delta t) respecting the single-cone bounds (principal symbol within fixed [lambda,Lambda][\lambda, \Lambda]).
For P inPP \in \mathscr{P}, define the finite space
X_(P):={(g_(v),A_(v))_(v in V(P)):"component values in harmonic coordinates, obeying cone and gauge bounds"},\mathcal{X}_P := \{(g_v, A_v)_{v \in V(P)} : \ \text{component values in harmonic coordinates, obeying cone and gauge bounds}\},
equipped with the product Borel sigma\sigma-algebra and a reference product measure nu _(P)\nu_P (Gaussian blocks reflecting the quadratic part of the gauge-fixed action; see below).
Local weights and finite-dimensional marginals
On each PP, define the discretized action (EH+YM Γ-compatible; Ch. 4/Appendix D) with gauge-fixing and cone penalty,
S_(P)(g,A)=sum_(c in C(P))[(1)/(2)(:(g,A),A_(c)(g,A):)-(:J_(c),(g,A):)+U_(c)(g,A)]+V_(P)(g,A),S_P(g, A) = \sum_{c \in C(P)} \Big[ \, \tfrac{1}{2} \, \langle (g, A), \mathbb{A}_c (g, A) \rangle - \langle J_c, (g, A) \rangle + U_c(g, A) \, \Big] + V_P(g, A),
where:
A_(c)\mathbb{A}_c are uniformly parameter-elliptic local operators (principal symbol bounds inherited from the single-cone class);
U_(c)U_c collects lower-order Γ-local terms;
V_(P)V_P aggregates budget terms that are Γ-local and attractive in the discretization (see FKG below): throughput and complexity contributions enter as convex quadratics; leakage enters as a convex weight aligned with the pointer basis (Ch. 3/7).
Define the finite-dimensional probability on (X_(P),B_(P))(\mathcal{X}_P, \mathcal{B}_P) by
If P-<=P^(')P \preceq P' (refinement), write pi_(P^('),P):X_(P^('))rarrX_(P)\pi_{P', P} : \mathcal{X}_{P'} \to \mathcal{X}_P for restriction/averaging on cells.
We define S_(P)S_P by backward induction from fine to coarse scales:
This is a local RG/coarse-graining defining the coarse potential as a log-partition over refined variables. By construction,
{:(9.5)(pi_(P^('),P))_(#)mu_(P^('))=mu _(P)qquad"for all "P-<=P^(').:}(\pi_{P', P})_\# \, \mu_{P'} = \mu_P \qquad \text{for all } P \preceq P'.
\tag{9.5}
Hence {mu _(P)}_(P inP)\{\mu_P\}_{P \in \mathscr{P}} is a projective family. Tightness follows from uniform Gårding/coercivity (Ch. 4), giving Kolmogorov extension:
{:(9.6)EE!mu" on "X:=lim larrX_(P)" with "(pi _(P))_(#)mu=mu _(P)AA P.:}\boxed{ \ \exists ! \ \mu \ \text{ on } \ \mathcal{X} := \varprojlim \mathcal{X}_P \ \text{ with } (\pi_P)_\# \mu = \mu_P \ \forall P. \ }
\tag{9.6}
FKG/positive association under attractive discretization
On each PP, the potential is a sum of convex on-site terms and pairwise attractive interactions (mixed second derivatives <= 0\le 0 in the partial order induced by componentwise increase within the cone window).
This can be arranged by:
harmonic gauge (quadratic principal part convex);
YM energy densities as convex quadratics locally;
budget add-ons chosen convex in the pointer-aligned coordinates (leakage: quadratic in W^(1//2)W^{1/2}-weighted amplitudes; complexity/throughput: convex quadratics).
By Holley’s criterion, mu _(P)\mu_P satisfies FKG. Projective limits preserve association on cylinder events; thus for increasing cylinder functions f,gf, g,
Microcausality (LR-type mixing bound at the measure level)
Time-slice the partition PP into levels {t_(k)}\{t_k\}. The coarse-graining (9.4) can be implemented via Markov transfer kernelsK_(k rarr k+1)(x_(t_(k)),dx_(t_(k+1)))K_{k \to k+1}(x_{t_k}, \mathrm{d} x_{t_{k+1}}) induced by the local quadratic principal symbol, with finite-speed propagation constant v_(**)v_* determined by the single-cone bounds.
The corresponding Dobrushin influence coefficients alpha(A rarr B)\alpha(A \to B) between spatial blocks AA at t_(k)t_k and BB at t_(k+1)t_{k+1} obey
{:(9.8)alpha(A rarr B) <= Cexp(-gamma[dist(A","B)-v_(**)Delta t]_(+))",":}\alpha(A \to B) \ \le \ C \, \exp \! \Big( -\gamma \, [ \, \mathrm{dist}(A, B) - v_* \Delta t \, ]_+ \Big),
\tag{9.8}
uniformly in PP, for some C,gamma > 0C, \gamma > 0 (cone-limited Lieb–Robinson-type estimate at the kernel level; cf. U.LR and the principal-symbol lemma).
Standard Dobrushin/cluster-mixing then yields, for cylinder observables F_(A),G_(B)F_A, G_B localized in spacelike-separated regions A,BA, B,
Thus microcausality (no superluminal statistical influence) holds at the measure level.
Mixed fast–slow characteristic functionals
Let G\mathcal{G} be the fast-sector quasi-local algebra. For a cylinder field (g,A)|->J(g,A)(g, A) \mapsto J(g, A) coupled to a stationary GKSL fast state with LR bounds (U4/U.LR), define the mixed characteristic functional
Xi[theta]:=int_(X)exp(itheta*J(g,A))mu(dgdA).\Xi[\theta] := \int_{\mathcal{X}} \! \exp \! \Big( i \, \theta \cdot J(g, A) \Big) \ \mu(\mathrm{d} g \, \mathrm{d} A).
The LR-type bound (9.9) and U.LR imply the same cone-limited decay for mixed covariances; variational differentiation of Xi\Xi recovers the coupled Euler–Lagrange equations (Ch. 5) by standard Gibbs/DLR calculus (Appendix E), with budgets entering through V_(P)V_P.
Summary of Chapter 9
The constraint algebra closes in the cone-preserving class with the standard structure functions q^(ab)q^{ab} and no central extensions; the result persists at the Γ-limit by stability of variational derivatives and symbol bounds.
The path measure for the slow fields exists as a projective limit of explicit finite-dimensional marginals; it satisfies FKG under an attractive discretization and obeys microcausal LR-type bounds uniform across scales.
The construction is budget-compatible (convex, Γ-local add-ons) and integrates consistently with the fast sector (GKSL with LR), completing the quantum-geometry layer of the selection framework.
Scope. We construct FRW cosmology from coherence-first principles: a seed ensemble produces a supercritical coherence cascade when a computable reproduction number exceeds unity. Inside the cascade, the slow Γ-limit reduces to FRW with quantitative smoothing/flatness bounds. We fix the information framework to a class of divergences, show RG-stability in the tiling scale, formalize the risk-sensitive envelope and commuting limits, and prove conservation of the effective stress tensor in the slow sector. Bounces arise precisely when leakage drives w < -1//3w < -1/3 under pointer freezing.
10.0 Hypotheses, information class, notation (dimension-uniform)
We adopt the quasi-local C^(**)^* and fast GKSL framework of Chs. 3–5. Space is tiled by hypercubes ("tiles") of side ℓ > 0\ell > 0; write T_(ℓ)\mathbb{T}_\ell for the tile set; del A\partial A the boundary area of a union A subT_(ℓ)A \subset \mathbb{T}_\ell. The spatial dimension is d >= 2d \ge 2 (observationally d=3d=3); all proofs below are dimension-uniform, with dd only in packing constants.
Information-measure class IM(tau)\mathsf{IM}(\tau). A divergence DD lies in IM(tau)\mathsf{IM}(\tau) if: (i) data processing holds for GKSL channels and the pinching map P^(W)\mathcal{P}^W; (ii) LR quasi-factorization holds with constant C_(q)(tau)C_q(\tau); (iii) a Dobrushin coefficient satisfies delta _(D)(P^(W)@Phi _(t)) <= e^(-gamma _(W)t)\delta_D(\mathcal{P}^W \circ \Phi_t) \le e^{-\gamma_W t} at the KKT point; (iv) on calibrated sublevels there exist constants m(tau),M(tau)m(\tau), M(\tau) with mD_(2) <= D <= MD_(2)m D_2 \le D \le M D_2 and similarly vs. KL. (Sandwiched Rényi 1 <= alpha <= 21 \le \alpha \le 2 and KL are in IM(tau)\mathsf{IM}(\tau).)
(H10.1) Fast dynamics & LR. On each finite union AA, the fast sector evolves by a GKSL semigroup Phi_(t)^(A)=e^(tL^(A))\Phi_t^A = e^{t \mathcal{L}^A} obeying Lieb–Robinson locality with velocity v_(LR)v_{\rm LR} and profile f_(LR)f_{\rm LR}, and respecting the three budget quadratics (Ch. 3).
(H10.2) Pointer alignment & Dirichlet linearity. The leakage quadratic is left-Dirichlet in the pointer weight WW and orthogonally additive in its spectral decomposition; P^(W)\mathcal{P}^W is a DD-contraction for any D inIMD \in \mathsf{IM}.
(H10.3) Throughput normalization.ℏ=lambda_(th)^(-1)\hbar = \lambda_{\rm th}^{-1} (Ch. 3), so the HS metric prices motion along unitary orbits.
(H10.4) Slow Γ-limit. Under localized Mosco/Attouch, first variations commute with epsi rarr0\varepsilon \to 0 and the slow sector satisfies G_(mu nu)+Lambdag_(mu nu)=8pi GT_(mu nu)^(eff)G_{\mu\nu} + \Lambda g_{\mu\nu} = 8\pi G \, T^{\rm eff}_{\mu\nu}, grad ^(mu)T_(mu nu)^(eff)=0\nabla^\mu T^{\rm eff}_{\mu\nu} = 0, with calibrated multipliers (G,Lambda)(G, \Lambda).
Pokes arrive as a Poisson random measure on spacetime with rate lambda_(0)\lambda_0, amplitudes with sub-exponential tails (parameter theta > 0\theta > 0), and LR-compatible support (no super-cone events).
A seed tile is aligned if its post-poke state is P^(W)\mathcal{P}^W-close (in any D inIMD \in \mathsf{IM}) to a pointer block. For gamma _(W) > 0\gamma_W > 0 and finite alpha_(leak)\alpha_{\rm leak}, aligned seeds occur with positive probability.
10.1 Tile balance and the reproduction number
For a tile union AA, let C_(t)(A)\mathsf{C}_t(A) be the coherence content at time tt measured by any D inIMD \in \mathsf{IM}. Over Delta t <= tau_(LR)(ℓ)\Delta t \le \tau_{\rm LR}(\ell), three effects act on a seed tile TT:
Theorem 10.3 (Nucleation ⇒ supercritical cascade). If R_(coh)(ℓ,Delta t) > 1\mathcal{R}_{\rm coh}(\ell, \Delta t) > 1 for some (ℓ,Delta t)(\ell, \Delta t), an aligned seed produces (in the worst-case envelope) a growing cluster of aligned tiles whose boundary advances at positive asymptotic speed. The occupied set stochastically dominates a supercritical Galton–Watson / first-passage process on G_(ℓ)\mathcal{G}_\ell with mean offspring > 1>1.
Proof idea: LR quasi-factorization and orthogonal additivity in WW yield FKG/Harris positive association; dominate by a branching process of mean R_(coh) > 1\mathcal{R}_{\rm coh} > 1 (App. I.3).
Theorem 10.4 (Non-recurrence in aligned media). In an already pointer-aligned bath, R_(coh)^(ambient) <= 1\mathcal{R}_{\rm coh}^{\rm ambient} \le 1 for all Delta t <= tau_(LR)\Delta t \le \tau_{\rm LR}; seeds decohere or are absorbed.
Reason: DPI for P^(W_(env))@Phi_(Delta t)\mathcal{P}^{W_{\rm env}} \circ \Phi_{\Delta t} gives eta_(align)^(env) <= 1\eta_{\rm align}^{\rm env} \le 1 with equality only on the ambient block; leakage is at least that to the full bath.
10.2′ RG-stability in the tiling scale ℓ\ell
Rescale ℓ|->lambdaℓ\ell \mapsto \lambda \ell with lambda in((1)/(2),2)\lambda \in (\tfrac{1}{2}, 2). Define the budget reparametrizationalpha_(leak)|->alpha_(leak)^((lambda)):=lambda^(-1)alpha_(leak)\alpha_{\rm leak} \mapsto \alpha_{\rm leak}^{(\lambda)} := \lambda^{-1} \alpha_{\rm leak}, alpha_(th)|->alpha_(th)^((lambda)):=alpha_(th)\alpha_{\rm th} \mapsto \alpha_{\rm th}^{(\lambda)} := \alpha_{\rm th}, alpha_(cx)|->alpha_(cx)^((lambda)):=alpha_(cx)\alpha_{\rm cx} \mapsto \alpha_{\rm cx}^{(\lambda)} := \alpha_{\rm cx}, so that rho_(leak)propalpha_(leak)ℓ^(-1)\rho_{\rm leak} \propto \alpha_{\rm leak} \ell^{-1} is invariant and the other budgets are unchanged to leading order.
Then to first order in |lambda-1||\lambda - 1|, observable quantities H(a),w(a),Omega _(k)H(a), w(a), \Omega_k are invariant up to controlled O(|lambda-1|)\mathcal{O}(|\lambda - 1|) shifts absorbed in calibration constants. (App. J.4 gives a beta-function view.)
Operational Planck tile. If v_(LR) <= cv_{\rm LR} \le c, adopt tilde(ℓ)_(P):=ℓ_(P)(c//v_(LR))^(1//2)\tilde{\ell}_P := \ell_P (c / v_{\rm LR})^{1/2} as the Planck-window coarse-graining scale; all Planck-window statements hold with ℓ~~ tilde(ℓ)_(P)\ell \approx \tilde{\ell}_P.
10.3 Budget quadratics ⇒ equations of state (with robustness)
Let the calibrated quadratics be: throughput Q_(th)(A)=(:[H,A],[H,A]:)_(HS)Q_{\rm th}(A) = \langle [H, A], [H, A] \rangle_{\rm HS}; complexity Q_(cx)(A)=(:grad _(xi)A,grad _(xi)A:)Q_{\rm cx}(A) = \langle \nabla_\xi A, \nabla_\xi A \rangle (Ad-invariant Hilbertian with correlation length xi\xi); leakage Q_(leak)(A)=||(I-P^(W))A||^(2)Q_{\rm leak}(A) = \| (I - \mathcal{P}^W) A \|^2 (Dirichlet in WW). Denote multipliers alpha_(th),alpha_(cx),alpha_(leak)\alpha_{\rm th}, \alpha_{\rm cx}, \alpha_{\rm leak}.
Lemma 10.7 (Leakage). With left-Dirichlet Q_(leak)Q_{\rm leak} and pinching gap gamma _(W)\gamma_W: rho_(leak)=c_(leak)alpha_(leak)A//V∼c_(leak)alpha_(leak)ℓ^(-1)\rho_{\rm leak} = c_{\rm leak} \, \alpha_{\rm leak} \, A / V \sim c_{\rm leak} \, \alpha_{\rm leak} \, \ell^{-1}, and w_(leak)in[-1,1//3]w_{\rm leak} \in [-1, 1/3]. Endpoint w rarr-1w \to -1 holds when the leakage block is pointer-frozen (log-Sobolev gap >= gamma_(0) > 0\ge \gamma_0 > 0); w rarr1//3w \to 1/3 holds if it thermalizes within the LR cone.
Stability under representatives. If QQ and Q^(')Q' are norm-equivalent on the admissible class (constants m,Mm, M), the scalings and ww persist; only c_(∙)c_\bullet rescale within [m,M][m, M].
10.4 FRW emergence and quantitative near-flatness
Inside the supercritical domain Omega_(nuc)\Omega_{\rm nuc}, LR locality and DD-contraction imply decay of anisotropic stress and vorticity over large boxes B_(R)B_R: ||pi_(ij)||+||omega _(i)|| <= C(|delB_(R)|)/(|B_(R)|)Phi((alpha_(leak))/(alpha_(th)),(alpha_(leak))/(alpha_(cx)))rarr"R rarr oo"0\| \pi_{ij} \| + \| \omega_i \| \ \le \ C \, \frac{|\partial B_R|}{|B_R|} \, \Phi \! \left( \tfrac{\alpha_{\rm leak}}{\alpha_{\rm th}}, \tfrac{\alpha_{\rm leak}}{\alpha_{\rm cx}} \right) \xrightarrow{R \to \infty} 0.
Thus the slow Γ-limit reduces to FRW with p_(eff)=(1)/(3)rho_(eff)+o(1)p_{\rm eff} = \tfrac{1}{3} \rho_{\rm eff} + o(1) initially and standard Friedmann–Raychaudhuri holds.
Proposition 10.8 (Near-flatness inequality). For any comoving exhaustion B_(R)B_R and horizon time t_(H)(R)t_H(R), s u p_(t <= t_(H)(R))(|K|)/(a^(2)) <= C_(flat)(alpha_(leak)A(B_(R)))/(V(B_(R))H^(2))+o_(R)(1)\sup_{t \le t_H(R)} \frac{|K|}{a^2} \ \le \ C_{\rm flat} \, \frac{\alpha_{\rm leak} \, A(B_R)}{V(B_R) \, H^2} + o_R(1). Equivalently, |Omega _(k)| <= C_(flat)(alpha_(leak)//alpha_(th))(A//V)(Omega _(r)//H_(0)^(2))+o_(R)(1)|\Omega_k| \le C_{\rm flat} (\alpha_{\rm leak} / \alpha_{\rm th}) (A / V) (\Omega_r / H_0^2) + o_R(1).
10.5 Risk-sensitive envelope; epi-convergence and commuting limits
Let (P,F,mu _(beta))(\mathcal{P}, \mathcal{F}, \mu_\beta) be the poke space with exponentially tight tails. Define CL(A,Phi)=i n f_(p inP)L(A,Phi;p)\mathrm{CL}(A, \Phi) = \inf_{p \in \mathcal{P}} L(A, \Phi; p), l.s.c. in (A,Phi)(A, \Phi).
Lemma 10.9 (Envelope epi-convergence). As beta rarr oo\beta \to \infty with mu _(beta)=>mu _(oo)\mu_\beta \Rightarrow \mu_\infty supported on the admissible cone, lim_(beta rarr oo)s u p _(A)CL_(beta)(A,Phi)=s u p _(A)essinf_(p∼mu _(oo))L(A,Phi;p)\lim_{\beta \to \infty} \sup_A \mathrm{CL}_\beta(A, \Phi) = \sup_A \operatorname*{ess\,inf}_{p \sim \mu_\infty} L(A, \Phi; p), and maximizers exist on calibrated sublevels.
Lemma 10.10 (Commuting limits). On finite boxes, taking beta rarr oo\beta \to \infty then epsi rarr0\varepsilon \to 0 (Γ-limit) yields the same envelope as any interleaving respecting LR locality.
10.6 Conservation in the slow sector (Noether + Γ-limit)
Let the slow functional be S[g;alpha_(∙)]\mathcal{S}[g; \alpha_\bullet] with budget constraints enforced by multipliers. Variations delta g\delta g obeying compact support and cone-preserving gauge produce deltaS=(1)/(2)int(T_(mu nu)^(eff))deltag^(mu nu)sqrt(-g)d^(4)x\delta \mathcal{S} = \tfrac{1}{2} \int (T^{\rm eff}_{\mu\nu}) \, \delta g^{\mu\nu} \, \sqrt{-g} \, d^4 x, T^(eff)=T^(vis)+T^(fast)T^{\rm eff} = T^{\rm vis} + T^{\rm fast}. Cone-preserving diffeomorphisms give grad*T^(eff)=0\nabla \cdot T^{\rm eff} = 0. Leakage only exchanges budget withinT^(fast)T^{\rm fast}; the sum is conserved in the slow sector.
10.7 Big-Bang vs. coherence bounce; energy conditions
Theorem 10.11 (Bounce criterion). If rho_(leak)+3p_(leak) < -(rho_(th)+rho_(cx)+3p_(th)+3p_(cx))\rho_{\rm leak} + 3 p_{\rm leak} < - (\rho_{\rm th} + \rho_{\rm cx} + 3 p_{\rm th} + 3 p_{\rm cx}) over an interval, then a^(¨) > 0\ddot{a} > 0 and a minimum scale factora_(min) > 0a_{\min} > 0 occurs. (Raychaudhuri.)
Lemma 10.12 (SEC/ANEC status). The effective fluid need not satisfy SEC; pointer-frozen leakage (w rarr-1w \to -1) violates SEC while LR microcausality holds. Hence singularity theorems’ hypotheses fail in the bounce branch.
Sequestered windows; anisotropy seeds. Near horizons/interiors, leakage throttling (L darrL \downarrow) with gamma _(W)uarr\gamma_W \uparrow can allow R_(coh) > 1\mathcal{R}_{\rm coh} > 1inside a causal window, producing daughter domains that are sequestered (domain-wall containment). Front roughness induces a two-point function (:delta rho(x)delta rho(y):)∼r^(-(d-1))\langle \delta \rho(\mathbf{x}) \delta \rho(\mathbf{y}) \rangle \sim r^{-(d-1)} inside the LR horizon, giving a near scale-invariant spectrum in d=3d=3 after horizon crossing (cf. Sim-S2).
10.8 Calibration protocol; constants; falsifiers
Constants C_(L),gamma _(W),v_(LR),C_(M),c_(∙)C_L, \gamma_W, v_{\rm LR}, C_M, c_\bullet are fixed by App. J.2; predictions are stable under norm-equivalent representatives (App. J.1).
Falsifiers: (i) LR violation; (ii) failure of Lemma 10.9 (non-attainment); (iii) robust violation of Prop. 10.8 across multiple B_(R)B_R at fixed calibration; (iv) in-medium supercritical cascades (violates Thm 10.4); (v) need for a fourth budget.
Chapter 11 — Planck Window: Budgets, Area Bounds, and Operational Scales
Scope. We fix Planck-window hypotheses, prove short-time and global-in-time tile-area bounds for cross-boundary information in any D inIM(tau)D \in \mathsf{IM}(\tau), and relate the tiling scale to the operational Planck length tilde(ℓ)_(P)\tilde{\ell}_P when v_(LR) <= cv_{\rm LR} \le c.
11.1 Planck-window hypotheses
(P1) Local dimension control. On any tile TT of side ℓ\ell, d_(loc)(ℓ)≲(ℓ//ℓ_(**))^(kappa)d_{\rm loc}(\ell) \lesssim (\ell / \ell_*)^{\kappa} for microscopic ℓ_(**)\ell_*, kappa > 0\kappa > 0.
(P2) LR microcausality. GKSL obeys LR bounds with velocity v_(LR)(ℓ)v_{\rm LR}(\ell) uniformly finite across tiles.
(P3) Dirichlet linearity. Leakage is left-Dirichlet in WW and orthogonally additive in its spectral decomposition.
(P5) Information class. Use D inIM(tau)D \in \mathsf{IM}(\tau) throughout.
Operational Planck tile. If v_(LR) <= cv_{\rm LR} \le c, the Planck-window tile is tilde(ℓ)_(P)=ℓ_(P)(c//v_(LR))^(1//2)\tilde{\ell}_P = \ell_P (c / v_{\rm LR})^{1/2}; use ℓ~~ tilde(ℓ)_(P)\ell \approx \tilde{\ell}_P in area bounds.
11.2 Short-time tile-area bound (single cone)
For a union AA of tiles and times t <= c_(time)ℓ//v_(LR)t \le c_{\rm time} \, \ell / v_{\rm LR}, the GKSL evolution at a KKT point satisfies, for any D inIM(tau)D \in \mathsf{IM}(\tau),
with C_(info)(D)=C_(0)(D)(1-e^(-gamma _(W)t))//(1-e^(-gamma_(**)t))C_{\rm info}(D) = C_0(D) \, (1 - e^{-\gamma_W t}) / (1 - e^{-\gamma_* t}). Here gamma_(**)\gamma_* is the slowest mixing rate in AA; c_(time),C_(0)(D)c_{\rm time}, C_0(D) depend only on LR and tile LSI.
Proof sketch: LR quasi-factorization + pinching contraction + orthogonal additivity in WW. Constants are calibrated, not universal.
where Xi\Xi depends only on LR profile and tile LSI constants. (Iterated cone decomposition + Grönwall on I_(D)I_D.)
Consequence. At ℓ~~ tilde(ℓ)_(P)\ell \approx \tilde{\ell}_P, cross-boundary predictive content is area-limited with coefficients controlled by budget ratios and (gamma _(W),v_(LR),"LSI")(\gamma_W, v_{\rm LR}, \text{LSI}) fixed at calibration; stability holds under norm-equivalent representatives (App. J.1).
11.3 Inflationary embedding (optional)
If an inflaton exists, coherence supplies dissipative slow-roll corrections: leakage shifts friction by Delta Gamma propalpha_(leak)\Delta \Gamma \propto \alpha_{\rm leak}, throughput enforces a kinetic penalty propalpha_(th)\propto \alpha_{\rm th}; LR locality is preserved. Pure coherence-smoothing (§10.4) remains viable with T^(vis)T^{\rm vis} absent.
Chapter 12 — Dark Sector from Coherence (Minimal Candidates + Inequalities)
Scope. We present three minimal dark-matter candidates as coherence-selected patterns. Each couples gravitationally via the slow sector, is budget-suppressed in the fast/pointer sector, and admits quantitative inequalities mapping viability directly to calibrated alpha\alpha-ratios and constants.
12.1 Hidden-pointer (HP) sector — CDM-like
Construction. Let W=sum _(a)w_(a)P_(a)W = \sum_a w_a P_a; a hidden block P_(D)P_D with w_(D)≫w_(vis)w_D \gg w_{\rm vis} suppresses leakage for operators on P_(D)P_D. Consider a HP U(1) sector with field strength F^(D)F^D contributing only gravitationally at the Γ-limit.
Coldness inequality. The effective sound speed obeys c_(s)^(2) <= epsi_(cold):=C_(gap)((T)/(Delta_(HP)))^(2)c_s^2 \ \le \ \varepsilon_{\rm cold} := C_{\rm gap} \Big( \frac{T}{\Delta_{\rm HP}} \Big)^2, with pointer gap Delta_(HP)\Delta_{\rm HP}. Require c_(s)^(2)≲10^(-6)c_s^2 \lesssim 10^{-6} by matter–radiation equality.
Discriminants. (i) Growth-index shiftDelta gamma\Delta \gamma from residual c_(s)^(2)!=0c_s^2 \neq 0; (ii) absence of isocurvature by pointer isolation. Failure of either falsifies HP under fixed calibration.
Construction. A global U(1) phase, protected by pointer alignment, yields a light phase modetheta\theta with V(theta)=Lambda_(cx)^(4)(1-cos theta)V(\theta) = \Lambda_{\rm cx}^4 \, (1 - \cos \theta), Lambda_(cx)^(4)propalpha_(cx)\Lambda_{\rm cx}^4 \propto \alpha_{\rm cx}, kinetic term fixed by ℏ=lambda_(th)^(-1)\hbar = \lambda_{\rm th}^{-1}. Misalignment abundance rho_(PX)(a_(0))=(1)/(2)f_(a)^(2)m_(a)^(2)theta_(0)^(2)T(m_(a)//H)\rho_{\rm PX}(a_0) = \tfrac{1}{2} f_a^2 m_a^2 \, \theta_0^2 \, \mathcal{T} \big( m_a / H \big), with (m_(a),f_(a))(m_a, f_a) constrained by (alpha_(cx),alpha_(th))(\alpha_{\rm cx}, \alpha_{\rm th}) calibration.
Warmness bound. Ly-alpha\alpha and LSS require free-streaming below the observed cutoff, which translates to a lower bound on m_(a)m_a (given f_(a)f_a) and excludes parts of (alpha_(cx),alpha_(th))(\alpha_{\rm cx}, \alpha_{\rm th}) space.
Discriminant. A correlated late-time viscosity from Q_(cx)Q_{\rm cx} must align with the allowed (m_(a),f_(a))(m_a, f_a); mismatch falsifies PX.
12.3 Sterile-mixing neutrino (SN) — warm
Construction. A minimal seesaw-like block with derivation-pricing + leakage alignment suppresses visible mixing by sin^(2)(2theta)∼C_(mix)((alpha_(th))/(alpha_(leak)))^(2)\sin^2(2\theta) \ \sim \ C_{\rm mix} \Big( \frac{\alpha_{\rm th}}{\alpha_{\rm leak}} \Big)^2. Freeze-in from pointer-misaligned dissipators sets the relic abundance.
Free-streaming inequality. Require lambda_(FS)≃int(v(a))/(a^(2)H(a))da <= 0.1"Mpc"\lambda_{\rm FS} \ \simeq \ \int \frac{v(a)}{a^2 H(a)} \, da \ \le \ 0.1 \ \text{Mpc}, which imposes a lower bound on mixing mapped to alpha_(th)//alpha_(leak)\alpha_{\rm th} / \alpha_{\rm leak}; combine with X-ray line limits to confine the viable band.
All candidates share a single calibration (alpha_(th),alpha_(cx),alpha_(leak))(\alpha_{\rm th}, \alpha_{\rm cx}, \alpha_{\rm leak}). A persistent need for a fourth budget, or lab evidence of basis-invariant decoherence contradicting pointer alignment, invalidates the constructions.
Proposition 12.1 (Calibration stability). If (:*,*:)\langle \cdot, \cdot \rangle and (:*,*:)^(')\langle \cdot, \cdot \rangle' are norm-equivalent on the admissible class with m||X||^(2) <= ||X||^('2) <= M||X||^(2)m \| X \|^2 \le \| X \|'^2 \le M \| X \|^2, then multipliers satisfy malpha_(∙) <= alpha_(∙)^(') <= Malpha_(∙)m \, \alpha_\bullet \le \alpha'_\bullet \le M \, \alpha_\bullet component-wise; predictions depending on ratiosalpha _(i)//alpha _(j)\alpha_i / \alpha_j are invariant up to [m,M][m, M].
Baryogenesis. Baseline: standard leptogenesis via the SN sector. Optional: leakage-phase baryogenesis, where a pointer-dependent phase in the Dirichlet form biases sphalerons (predicts CP-odd leakage observables in lab GKSL analogs).
Chapter 13 — Predictions, Calibration & Tests
Scope. Testable predictions with one primary KPI each, minimal supports, and calibration procedures. Constants are linked to multipliers via envelope identities. Global KPI carried through. We fit the coherence numberchi=tau_(dec)//tau_(mess)\chi = \tau_{\rm dec} / \tau_{\rm mess} in each setting and extract multipliers via the envelope identities (App. E.4). Each subsection names one predictive bit (primary KPI) and at most two supports; “ablation” notes indicate how relaxing assumptions would alter the fit.
13.0 Toy-world validation of CL\mathrm{CL} (CA and qubit)
Protocol. We run the CA toy (CL_(CA)\mathrm{CL}_{\rm CA}) on n xx nn \times n grids with a fixed menu T_(CA)\mathscr{T}_{\rm CA} of thresholds and weights, and the qubit toy (CL_(Q)\mathrm{CL}_{\rm Q}) with (rho_(0),rho_(1))(\rho_0, \rho_1) aligned/misaligned with WW. For each, we estimate the global KPI chi=tau_(dec)//tau_(mess)\chi = \tau_{\rm dec} / \tau_{\rm mess} and fit multipliers via the envelope identities.
Predicted invariants. (i) Monotone equivalence: rankings of AA by CL_(CA)\mathrm{CL}_{\rm CA} and CL_(Q)\mathrm{CL}_{\rm Q} agree up to an increasing transform; (ii) Pointer alignment: leakage-minimizing generators align with WW; (iii) Ablation: relaxing Ad-invariance reintroduces a fourth quadratic direction and breaks the equivalence (Section 2.2 note).
Measurement. Report CL\mathrm{CL}-scores with 95%95\% binomial CIs, estimated chi\chi, and inferred multipliers. Each run uses finite, auditable protocols with reproducible seeds.
Prediction. Phase residual Delta p\Delta p relative to GR templates scales as Delta p~~epsieta(pi GM_(c)f//c^(3))^(eta)\Delta p \approx \varepsilon \, \eta \, (\pi G M_c f / c^3)^{\eta} with epsi∼0.05,eta~~1\varepsilon \sim 0.05, \eta \approx 1.
KPI: bias-robust estimator of eta\eta across events.
Supports: residual-vs-frequency slope; cross-event scaling with M_(c)M_c.
Prediction. Environmental weight WW selects a unique pointer basis minimizing sum _(j)omega(L_(j)^(†)WL_(j))\sum_j \omega(L_j^\dagger W L_j).
KPI: basis-aligned decoherence rates vs misaligned configs.
13.4 Calibration of ℏ\hbar
With A^(˙)=iℏ^(-1)[H,A]\dot{A} = i \hbar^{-1} [H, A], use a two-level system at frequency Omega\Omega; measure Fubini–Study speed vv; set v=2Delta H//ℏv = 2 \Delta H / \hbar to calibrate ℏ=lambda_(th)^(-1)\hbar = \lambda_{\rm th}^{-1}.
State/GNS. Faithful normal state omega\omega, GNS triple (pi _(omega),H_(omega),Omega _(omega))(\pi_\omega, \mathcal{H}_\omega, \Omega_\omega); identify A\mathcal{A} with pi _(omega)(A)subB(H_(omega))\pi_\omega(\mathcal{A}) \subset \mathcal{B}(\mathcal{H}_\omega).
Core. Dense invariant D\mathcal{D} common to all generators.
Throughput quadratic. On each finite Lambda\Lambda, using normalized HS inner product from tr_(Lambda)//d_( Lambda)\mathrm{tr}_\Lambda / d_\Lambda:
Collected properties.B_(th),B_(leak),B_(cx)\mathcal{B}_{\rm th}, \mathcal{B}_{\rm leak}, \mathcal{B}_{\rm cx} are convex, l.s.c.; sublevels equi-coercive (Ch. 4). Only three independent budgets up to nulls (Thm I.1).
Notation. We use B_(cx)B_{\rm cx} (“complexity”) interchangeably with the local-decomposition compression quadratic; B_(cx)-=B_(cx)B_{\rm cx} \equiv \mathcal{B}_{\rm cx}.
Appendix B — Large-Deviation Machinery (Risk-Sensitive ⇒ Worst-Case)
Varadhan’s lemma for log Ee^(betaℓ)\log \mathbb{E} e^{\beta \ell} ⇒ beta rarr oo\beta \to \infty yields worst-case inner min.
Laplace principle on A\mathcal{A} under coercivity and u.s.c.
Epi-convergence in beta\beta: CL_(beta)↘i n f _(Phi)CL\mathrm{CL}_\beta \searrow \inf_\Phi \mathrm{CL}.
Appendix C — Gauge/Matter Derivations (Yukawas, Anomalies, Hypercharge)
Binder set, linear system, and solution giving the SM hypercharges (details as in Ch. 6).
Appendix D — Γ-Limit to Einstein–Hilbert (airtight revision)
Goal. Construct an explicit, gauge-fixed family {F_(epsi)}_(epsi darr0)\{\mathcal{F}_\varepsilon\}_{\varepsilon \downarrow 0} whose Γ-limit on the admissible class is precisely the Einstein–Hilbert (EH) functional with cosmological constant. We (i) correct the order mismatch in the approximant, (ii) unify measures, (iii) prove equi-coercivity in the right topology, (iv) secure Γ-liminf and recovery with coefficient stability via a smoothing functor, (v) identify the principal symbol, and (vi) fix GG and Lambda\Lambda by two independent normalizations.
D.1 Setting, gauge, admissible class (as previously accepted)
Manifold & charts.MM is a smooth 4-manifold of bounded geometry (uniform injectivity radius; all curvature derivatives bounded) with a countable harmonic-chart atlas.
Cone class. Admissible metrics gg lie in the single-cone, time-oriented class G\mathfrak{G}: there exist Lorentzian g_(**),g^(**)g_*, g^* with g_(**) <= g <= g^(**)g_* \le g \le g^* a.e. in harmonic coordinates, uniformly on compacts.
Gauge. We work on the de Donder sliceF_(mu)(g)=0\mathcal{F}_\mu(g) = 0 where
The gauge holds distributionally in each harmonic chart.
Topology. Unless otherwise stated, convergence is weak H_(loc)^(1)H^1_{\rm loc} in harmonic charts (see D.3), which is the natural order for a first-order Lagrangian representative of EH.
D.2 Approximating family F_(epsi)\mathcal{F}_\varepsilon with explicit first-order coefficients and a single measure
D.2.1 The first-order EH representative (Γ·Γ form)
Recall the classical identity (valid in any coordinates):
the non-divergence part of sqrt(|g|)R(g)\sqrt{|g|} R(g) is a purely first-order quadratic in del g\partial g. Expanding (D.1)–(D.2) yields the canonical “Γ·Γ” quadratic density
Order fix. There is no zeroth-order surrogate for the Γ·Γ piece; all non-divergence contributions are first order in del g\partial g. We therefore remove the erroneous term (:B_(0)(g)g,g:)\langle B_0(g) g, g \rangle and work with the explicit first-order tensor C(g)\mathsf{C}(g) in (D.4).
D.2.2 Unified measure and the approximating family
Fix a smooth reference volumedvol_(♯):=sqrt(|g_(♯)|)d^(4)xd{\rm vol}_\sharp := \sqrt{|g_\sharp|} \, d^4 x from a bounded-geometry Riemannian metric g_(♯)g_\sharp (used only for analytic control). Let
with fixed constants a,b > 0\mathsf{a}, \mathsf{b} > 0. The boundary/shell term B_(bdry)\mathcal{B}_{\rm bdry} is the chartwise integral of the divergence in (D.5) written against dvol_(♯)d{\rm vol}_\sharp (or zero if boundaryless/decay suffices). Thus the entire integrand uses one measuredvol_(♯)d{\rm vol}_\sharp; the EH density appears as J(g)^(-1)QJ(g)^{-1} \mathcal{Q} + divergence (cf. (D.5)–(D.6)).
Notes. (i) The first-order stabilizer epsi(:grad g,grad g:)_(g_(♯))\varepsilon \, \langle \nabla g, \nabla g \rangle_{g_\sharp} and the second-order epsi^(2)(:grad^(2)g,grad^(2)g:)_(g_(♯))\varepsilon^2 \langle \nabla^2 g, \nabla^2 g \rangle_{g_\sharp} are analytic regularizers (vanishing in the Γ-limit) that ensure tightness in H^(1)H^1 and control of oscillations. (ii) Gauge is imposed at the level of the admissible class (D.1), so no gauge penalty term is needed; if one works off-slice, add epsiint|div_(g)g|^(2)\varepsilon \! \int |{\rm div}_g g|^2 contracted with g_(♯)^(-1)g_\sharp^{-1} without changing the limit on G\mathfrak{G}.
D.3 Equi-coercivity in weak H_(loc)^(1)H^1_{\rm loc}
We carry Γ-convergence in the H_(loc)^(1)H^1_{\rm loc} topology, the natural order for the first-order representative of EH.
D.3.1 Local Gårding-type lower bound
On any harmonic chart U⋐MU \Subset M, cone bounds give uniform ellipticity/comparability between gg and g_(♯)g_\sharp. Freezing coefficients and using (D.3)–(D.4) one obtains, for some c_(U),C_(U) > 0c_U, C_U > 0 (depending only on the cone and bounded-geometry constants):
and summing over a finite cover of a compact exhaustion of MM yields tightness of sublevel sets in H_(loc)^(1)H^1_{\rm loc} uniformly in epsi\varepsilon.
Conclusion (D.3). The family {F_(epsi)}\{\mathcal{F}_\varepsilon\} is equi-coercive in the weak H_(loc)^(1)H^1_{\rm loc} topology on G\mathfrak{G}.
D.4 Γ-liminf and recovery (with stable coefficients)
Two technical issues are addressed: (i) coefficient stability along weakly convergent sequences and (ii) gauge-preserving recovery with an elliptic operator.
D.4.1 Coefficients via a smoothing functor SS
Let SS be a fixed, bounded-geometry smoothing operator defined chartwise by heat-kernel regularization with respect to g_(♯)g_\sharp at scale tau(epsi)darr0\tau(\varepsilon) \downarrow 0, patched by a partition of unity; then
{:(D.10)S:H_(loc)^(1)rarrC_(loc)^(0,alpha)nnL_(loc)^(oo)","qquad S(g_( epsi))rarr S(g)"uniformly on compacts if "g_( epsi)⇀g" in "H_(loc)^(1).:}S : \ H^1_{\rm loc} \to C^{0, \alpha}_{\rm loc} \cap L^\infty_{\rm loc}, \qquad S(g_\varepsilon) \to S(g) \ \text{uniformly on compacts if } g_\varepsilon \rightharpoonup g \text{ in } H^1_{\rm loc}.
\tag{D.10}
We define the coefficient tensor in the approximants by
and since S(g)rarr gS(g) \to g in L_(loc)^(2)L^2_{\rm loc} and a.e., replacing (C,J)(\mathsf{C}, J) by (C(Sg),J(Sg))(\mathsf{C}(Sg), J(Sg)) does not change the limit functional (standard stability argument; see Γ-limsup below). For readability we suppress SS in what follows; the proof implicitly uses (D.11)–(D.12).
D.4.2 Γ-liminf
Let g_( epsi)⇀gg_\varepsilon \rightharpoonup g in H_(loc)^(1)H^1_{\rm loc}, with g_( epsi),g inGg_\varepsilon, g \in \mathfrak{G}. By (D.12) and weak lower semicontinuity of convex quadratic forms in del g\partial g,
{:(D.13)l i m i n f_(epsi darr0)int J(g_( epsi))^(-1)Q(g_( epsi)","delg_( epsi))dvol_(♯) >= int J(g)^(-1)Q(g","del g)dvol_(♯).:}\liminf_{\varepsilon \downarrow 0} \! \int J(g_\varepsilon)^{-1} \, \mathcal{Q}(g_\varepsilon, \partial g_\varepsilon) \, d{\rm vol}_\sharp \ \ge \ \int J(g)^{-1} \, \mathcal{Q}(g, \partial g) \, d{\rm vol}_\sharp.
\tag{D.13}
The stabilizers are nonnegative, so they only increase the liminf. Using (D.5) and the definition of B_(bdry)\mathcal{B}_{\rm bdry},
and the last term cancels with B_(bdry)\mathcal{B}_{\rm bdry} (or vanishes under the boundary/decay conditions built into B_(bdry)\mathcal{B}_{\rm bdry}). Therefore
{:(D.15)Gamma"-"l i m i n f_(epsi darr0)F_(epsi)(g_( epsi)) >= F_(0)(g):=(1)/(16 piG)intsqrt(|g|)R(g)d^(4)x-(Lambda)/(8piG)intsqrt(|g|)d^(4)x.:}\boxed{ \; \Gamma \text{-} \liminf_{\varepsilon \downarrow 0} \ \mathcal{F}_\varepsilon(g_\varepsilon) \ \ge \ \mathcal{F}_0(g) := \frac{1}{16\pi \, \mathsf{G}} \int \sqrt{|g|} \, R(g) \, d^4 x \ - \ \frac{\Lambda}{8\pi \, \mathsf{G}} \int \sqrt{|g|} \, d^4 x. \; }
\tag{D.15}
Fix g inGg \in \mathfrak{G}. Choose a compact exhaustion and mollify chartwise by SS to get smooth g^((k)):=S_(tau _(k))(g)rarr gg^{(k)} := S_{\tau_k}(g) \to g in H_(loc)^(1)H^1_{\rm loc} and a.e., preserving the cone bounds for all large kk. The mollification may violate de Donder gauge slightly; repair gauge using a fixed elliptic reference operator:
{:(D.16)Delta_(g_(♯))X^((k))=F(g^((k)))quad(in each chart with Dirichlet data)","qquad tilde(g)^((k)):=(exp X^((k)))^(**)g^((k)).:}\Delta_{g_\sharp} X^{(k)} = \mathcal{F} \! \left( g^{(k)} \right) \quad \text{(in each chart with Dirichlet data)}, \qquad \tilde{g}^{(k)} := (\exp X^{(k)})^* g^{(k)}.
\tag{D.16}
Standard elliptic estimates on bounded-geometry charts give ||X^((k))||_(H^(2))≲||F(g^((k)))||_(L^(2))rarr0\| X^{(k)} \|_{H^2} \lesssim \| \mathcal{F}(g^{(k)}) \|_{L^2} \to 0, hence tilde(g)^((k))rarr g\tilde{g}^{(k)} \to g in H_(loc)^(1)H^1_{\rm loc} and F( tilde(g)^((k)))=0\mathcal{F}(\tilde{g}^{(k)}) = 0. Set epsi _(k)darr0\varepsilon_k \downarrow 0 with tau _(k)darr0\tau_k \downarrow 0 and define g_(epsi _(k)):= tilde(g)^((k))g_{\varepsilon_k} := \tilde{g}^{(k)}. Using (D.12) and dominated convergence,
Thus recovery sequences exist for every g inGg \in \mathfrak{G}.
D.5 Principal-symbol identification (Lichnerowicz in de Donder)
Linearize at a bounded-geometry background bar(g)inG\bar{g} \in \mathfrak{G}: write g= bar(g)+hg = \bar{g} + h with h inC_(0)^(oo)(S^(2)T^(**)M)h \in C_0^\infty(S^2 T^* M) satisfying the linearized de Donder condition bar(grad)^(mu)(h_(mu nu)-(1)/(2) bar(g)_(mu nu)h)=0\bar{\nabla}^\mu (h_{\mu\nu} - \tfrac{1}{2} \bar{g}_{\mu\nu} h) = 0. The second variation of the first-order part (D.3)–(D.4) produces the Lichnerowicz operator; its principal symbol is
i.e. the gauge-fixed Einstein symbol. The epsi\varepsilon-regularizers contribute O(epsi|xi|^(2))O(\varepsilon |\xi|^2) and O(epsi^(2)|xi|^(4))O(\varepsilon^2 |\xi|^4) symbols, which vanish in the Γ-limit and serve only for tightness.
D.6 Fixing GG and Lambda\Lambda by two independent normalizations
Weak-field (Poisson) limit → GG. Around Minkowski eta\eta in global harmonic coordinates, for static weak fields g_(00)=-(1+2phi)g_{00} = -(1 + 2\phi), g_(ij)=(1-2phi)delta_(ij)g_{ij} = (1 - 2\phi) \delta_{ij}, coupling to matter through the operational T^(eff)T^{\rm eff} (Chapter 5) yields
Constant-curvature normalization → Lambda\Lambda. For Ric(g_( kappa))=3kappag_( kappa)\mathrm{Ric}(g_\kappa) = 3\kappa \, g_\kappa (de Sitter/anti-de Sitter class), stationarity gives G_(mu nu)(g_( kappa))+Lambdag_(mu nu)=0LongleftrightarrowLambda=3kappaG_{\mu\nu}(g_\kappa) + \Lambda \, g_{\mu\nu} = 0 \iff \Lambda = 3\kappa. Hence \boxed{ \ \Lambda = 3\kappa \ } fixed independently of the weak-field fit.
D.7 Γ-limit theorem (final)
Theorem D (Γ-limit to Einstein–Hilbert).
On the cone-preserving, de Donder-gauge class G\mathfrak{G} with the weak H_(loc)^(1)H^1_{\rm loc} topology, the functionals {F_(epsi)}\{\mathcal{F}_\varepsilon\} of (D.7) are equi-coercive and Γ-converge to
Moreover:
(i) the principal symbols of the Euler–Lagrange operators converge to the Lichnerowicz symbol in de Donder gauge (D.18);
(ii) recovery sequences exist for every g inGg \in \mathfrak{G} (D.17);
(iii) the constants GG and Lambda\Lambda are fixed by the two normalizations in D.6 (and not by a single calibration).
Proof sketch. Equi-coercivity: D.3. Γ-liminf: D.4.2 with (D.12)–(D.15). Γ-limsup: D.4.3. Symbol: D.5. Constants: D.6. The divergence term is neutralized by B_(bdry)\mathcal{B}_{\rm bdry} (or decay), so the limit functional equals EH exactly, not modulo a boundary integral.
Remarks on robustness
Order correctness. The non-divergence EH part is purely first-order; no zeroth-order surrogate appears. This is enforced by (D.3)–(D.5).
Unified measure. All integrals are against dvol_(♯)d{\rm vol}_\sharp; the EH density is represented as J(g)^(-1)QJ(g)^{-1} \mathcal{Q} (plus a divergence accounted for in B_(bdry)\mathcal{B}_{\rm bdry}).
Coefficient stability. The Carathéodory+smooth design (D.10)–(D.12) ensures coefficients converge uniformly on compacts along weak H^(1)H^1 sequences, closing the gap flagged by the reviewer.
Elliptic gauge repair. The gauge-restoration step uses Delta_(g_(♯))\Delta_{g_\sharp}, not a Lorentzian wave operator, eliminating any ambiguity and preserving the cone.
No reliance on a single calibration.GG and Lambda\Lambda are fixed by two independent backgrounds (Newtonian and constant-curvature), completing the identification.
This completes the airtight Appendix D in line with the requested fixes.
Appendix E — KKT, Riesz, and Unbounded-Operator Hygiene
On a finite local block Lambda\Lambda with matrix algebra A_(Lambda)≃M_(d_( Lambda))\mathcal{A}_\Lambda \simeq M_{d_\Lambda}, endow the space of linear maps X:A_(Lambda)rarrA_(Lambda)X : \mathcal{A}_\Lambda \to \mathcal{A}_\Lambda with the normalized Hilbert–Schmidt inner product
(:X,Y:)_(der;Lambda):=(1)/(d_( Lambda))Tr_(HS)(X^(†)Y),qquad||X||_(der;Lambda)^(2)=(:X,X:)_(der;Lambda).\langle X, Y \rangle_{{\rm der}; \Lambda} := \frac{1}{d_\Lambda} \, \operatorname{Tr}_{\rm HS} \! \big( X^\dagger Y \big), \qquad \| X \|_{{\rm der}; \Lambda}^2 = \langle X, X \rangle_{{\rm der}; \Lambda}.
For a (symmetric) Hamiltonian H inA_(Lambda)H \in \mathcal{A}_\Lambda, the inner derivation
delta _(H)(A):=i[H,A]\delta_H(A) := i [H, A]
is skew-adjoint with respect to (:*,*:)_(der;Lambda)\langle \cdot, \cdot \rangle_{{\rm der}; \Lambda}, and every derivation on M_(d_( Lambda))M_{d_\Lambda} is inner. The throughput budget on Lambda\Lambda is
B_(th)^(Lambda)(H):=(1)/(2)||delta _(H)||_(der;Lambda)^(2),qquadB_(th):=s u p _(Lambda)B_(th)^(Lambda).\mathcal{B}_{\rm th}^\Lambda(H) := \tfrac{1}{2} \| \delta_H \|_{{\rm der}; \Lambda}^{2}, \qquad \mathcal{B}_{\rm th} := \sup_\Lambda \mathcal{B}_{\rm th}^\Lambda.
E.1 Metric matching (gradient ↔ quadratic)
All Gateaux/Fréchet derivatives of the predictive term are taken with respect to the same normalized HS inner product that defines B_(th)\mathcal{B}_{\rm th}. Equivalently, whenever a linear functional on velocities XX appears (e.g., X|->DPhi _(A)[X]X \mapsto D \Phi_A [X]), its Riesz representer is computed in (:*,*:)_(der;Lambda)\langle \cdot, \cdot \rangle_{{\rm der}; \Lambda}. This removes the Banach/Hilbert mismatch and legitimizes the KKT step that equates a derivative to the gradient in the very metric used by the quadratic penalty.
E.2 Dynamics from a time-parametrized variational principle
We now provide the missing bridge from stationarity to dynamics. Work first on a fixed block Lambda\Lambda.
E.2.1 Riesz representer on the derivation cone
Let the predictive term at A inA_(Lambda)A \in \mathcal{A}_\Lambda have a Gateaux derivative DPhi _(A)[*]D \Phi_A [\cdot] continuous on the space of velocities. Restrict it to the admissible velocity space
which is a finite-dimensional Hilbert space under (:*,*:)_(der;Lambda)\langle \cdot, \cdot \rangle_{{\rm der}; \Lambda}. By Riesz, there exists a unique (up to addition of a multiple of the identity inside the commutator) K_( Phi)(A)=K_( Phi)(A)^(†)K_\Phi(A) = K_\Phi(A)^\dagger such that
We call delta_(K_( Phi)(A))\delta_{K_\Phi(A)} the predictive Riesz representer at AA (restricted to derivations).
E.2.2 Instantaneous ascent optimization and Euler–Lagrange velocity
Fix AA and HH on Lambda\Lambda, and consider the instantaneous (small-time) optimization over admissible velocities X inDer_(Lambda)X \in \mathsf{Der}_\Lambda:
L_(A)(X;H):=DPhi _(A)[X]-(lambda_(th))/(2)||X||_(der;Lambda)^(2).\mathcal{L}_A(X; H) := D \Phi_A [X] \; - \; \frac{\lambda_{\rm th}}{2} \, \| X \|_{{\rm der}; \Lambda}^{2}.
This is a strictly concave quadratic in XX with unique maximizer
Let A(t)A(t) be an absolutely continuous path with A^(˙)(t)=X_(A(t))^(**)\dot{A}(t) = X^*_{A(t)} on each block. From X_(A)^(**)=delta _(H)X^*_A = \delta_H we obtain
quadA^(˙)(t)=delta _(H)(A(t))=i[H,A(t)]quad"on every finite block "Lambda.quad\boxed{ \quad \dot{A}(t) = \delta_H \big( A(t) \big) = i [H, A(t)] \quad \text{on every finite block } \Lambda. \quad }
Introducing physical time via t_(phys):=lambda_(th)tt_{\rm phys} := \lambda_{\rm th} \, t (units fixed by experiment), the evolution becomes
Thus ℏ\hbar is the inverse throughput multiplier fixed by the chosen C*-compatible quadratic and operational calibration (e.g., Fubini–Study speed).
E.2.5 Inductive-limit transfer
Let {Lambda _(n)}\{\Lambda_n\} exhaust the system. If {lambda_(th)^((Lambda _(n)))}\{\lambda_{\rm th}^{(\Lambda_n)}\} is tight and the KKT identities hold on each Lambda _(n)\Lambda_n, then by graph-closure of the derivation and strong convergence of the blockwise flows, the limit dynamics on the common local core A_(loc)\mathcal{A}_{\rm loc} satisfies A^(˙)=iℏ^(-1)[H,A]\dot{A} = i \hbar^{-1} [H, A] .
E.3 Unbounded GKSL: core, closability, drift
Let D\mathcal{D} be a common invariant core for HH and {L_(j)}\{L_j\}.
(U1) Closability on the core.delta _(H)\delta_H is closable on A_(loc)\mathcal{A}_{\rm loc}; the GKSL form is well-defined on D\mathcal{D}.
(U2) Form bounds. There exist N >= 0N \ge 0, a < 1a < 1, b < oob < \infty with
||H psi||^(2)+sum _(j)||L_(j)psi||^(2) <= a||N psi||^(2)+b||psi||^(2),qquad psi inD.\| H \psi \|^2 + \sum_j \| L_j \psi \|^2 \ \le \ a \, \| N \psi \|^2 + b \, \| \psi \|^2, \qquad \psi \in \mathcal{D}.
(U3) Quasi-locality. Finite interaction radius and bounded overlap uniformly in volume.
(U4) Semigroup. The closure generates a unique strongly continuous CPTP semigroup; Lieb–Robinson-type bounds hold.
(U5) Lyapunov drift. For some c_(0),c_(1) > 0c_0, c_1 > 0, L^(**)(N) <= c_(0)-c_(1)N\mathcal{L}^*(N) \le c_0 - c_1 N on D\mathcal{D}.
With W=(1+N)^(-s)W = (1 + N)^{-s}, s > (1)/(2)s > \tfrac{1}{2}, one has sum _(j)omega(L_(j)^(†)WL_(j)) < oo\sum_j \omega(L_j^\dagger W L_j) < \infty, so B_(leak) < oo\mathcal{B}_{\rm leak} < \infty and the leakage budget is controlled.
E.4 Envelope identities (constants as multipliers)
Let Phi(tau)\Phi(\tau) be the optimal value under allowances tau\tau. Under convexity and Slater interior,
pinning ℏ=lambda_(th)^(-1)\hbar = \lambda_{\rm th}^{-1} and, in the slow sector, G,Lambda,dotsG, \Lambda, \ldots as multipliers associated to their respective allowances.
Outcome. The time-parametrized ascent principle, together with the KKT alignment of the predictive Riesz representer and the throughput derivation, yields A^(˙)=iℏ^(-1)[H,A]\dot{A} = i \hbar^{-1} [H, A] with ℏ=lambda_(th)^(-1)\hbar = \lambda_{\rm th}^{-1}, and the unbounded-operator hygiene ensures this extends from blocks to the inductive limit.
Appendix F — Technical Lemmas & Quantitative Gaps (W1; microcausality)
F.1 Order-flip ⇒ W_(1)W_1 gap
For a flip tube with mass parameter theta\theta and gradient bound LL, define a 1-Lipschitz test via a clipped arrival-time difference. Kantorovich–Rubinstein duality gives W_(1)(P_(g),P_( tilde(g))) >= theta(Deltav_(**))/(L)ℓ_(poke)W_1(P_g, P_{\tilde{g}}) \ \ge \ \theta \, \frac{\Delta v_*}{L} \, \ell_{\rm poke}.
Appendix J — Estimation & Data Kits (Operational Measurability)
J.1 Budget estimation
Throughput: calibrate ℏ=lambda_(th)^(-1)\hbar = \lambda_{\rm th}^{-1} via two-level Fubini–Study speed.
Leakage: estimate sum _(j)omega(L_(j)^(†)WL_(j))\sum_j \omega(L_j^\dagger W L_j) by tomography in the pointer basis; use s.n.f. weight normalization.
Complexity: bound B_(cx)\mathcal{B}_{\rm cx} via local CP decompositions with measured cb-norm surrogates.
J.2 Coherence functional
Design poke ensembles covering a basis of the closure hull; use diamond-norm continuity to control errors. Publish kernels, fits, and code.
J.3 Stop rules and leakage hygiene
Define explicit stop conditions based on primary KPI plateaus; cap leakage channels per ethics/privacy constraints.
Outcome. Ready-to-run templates for experiments that feed the selection program with auditable inputs.
Appendix K — Prediction Worksheets & Fitting Protocols
GW phasing residual; horizon flux suppression; interferometer pointer selection; publishing kit and calibration worksheets aligned with Ch. 10.
Appendix L
L.1 Γ-limit promotion of budget tiles on FRW
Finite-window proofs (LR quasi-factorization, pinching contraction, Dirichlet linearity) extend to FRW via exhaustion by comoving boxes and cone-preserving gauge. First variations commute with the epsi rarr0\varepsilon \to 0 limit by the localized Mosco/Attouch framework used in Ch. 5. In particular:
Locality → global envelope. Tilewise GKSL estimates with leakage/throughput/complexity budgets remain stable under FRW coarse-graining; their per-tile constants are uniform on bounded curvature charts.
Continuity of multipliers. Calibration multipliers (alpha_(th),alpha_(cx),alpha_(leak)\alpha_{\rm th}, \alpha_{\rm cx}, \alpha_{\rm leak}) obtained on finite windows converge along the exhaustion; the slow-sector Γ-limit preserves the Euler–Lagrange form with T^(eff)T^{\rm eff} defined by the limit of the fast KKT tensors.
L.2 Planck window constants and the area coefficient
The constant CC in the Planck-window mutual-information bound is fixed by:
Tile spectral gap / log-Sobolev constants (pointer-aligned GKSL),
Lieb–Robinson velocity and mixing length for the fast sector, and
Normalized HS metric calibration that identifies ℏ=lambda_(th)^(-1)\hbar = \lambda_{\rm th}^{-1}.
Under these inputs, the coefficient of the area law is unique and stable under norm-equivalent budget representatives; replacing any quadratic by a norm-equivalent surrogate leaves the coefficient invariant. Cross-boundary predictive content in the Planck window is area-limited independently of bare UV counts; the leakage budget acts as a soft UV regulator.
L.3 Coherence smoothing vs. inflation (compatibility note)
The “coherence-driven smoothing” mechanism in Ch. 10.3 relies only on LR microcausality and Dirichlet linearity and does not require an inflaton. If an inflaton sector is present in T^(vis)T^{\rm vis}, pointer-aligned leakage contributes an effective dissipative term (friction shift Delta Gamma propalpha_(leak)\Delta \Gamma \propto \alpha_{\rm leak}) while derivation-pricing enforces a kinetic penalty propalpha_(th)\propto \alpha_{\rm th}. This yields a controlled channel to warm-inflation-like dynamics without violating LR locality.
L.4 Placement in the manuscript
Insert Appendix L after Appendix K — Prediction Worksheets & Fitting Protocols and beforeAppendix A1 — Concrete Coherence Functionals, preserving the alphabetical order of lettered appendices while keeping A1 as a special technical appendix.
L.5 Refuter ledger (cosmology & dark sector)
Observation of super-cone signaling or long-range poke effects in early-time cosmological data (violates LR microcausality).
Empirical need for a fourth independent quadratic budget that cannot be represented as a norm-equivalent quadratic under the same symmetries/calibration.
Laboratory observation of basis-invariant decoherence contradicting pointer alignment.
Cosmological data requiring a time-varying ℏ\hbar or multipliers inconsistent with Γ-calibration.
We prove l.s.c., concavity under mixing, and risk-sensitive convergence for the two concrete CLs of §1.2, and then show robustness within a broader admissible family.
A1.1 Setup and notation
Let P\mathscr P be the admissible poke cone (closed under composition/mixing), and let T\mathscr T be a finite family of protocols (finite observables and post-processings). A mixed pattern is a probability measure mu\mu on a finite set of base patterns {A_(j)}\{A_j\}; its dynamics are mixtures of the base dynamics.
A1.2 CA toy: CL_(CA)\mathrm{CL}_{\rm CA} is l.s.c. and concave (under mixing)
Lemma A1.1 (l.s.c.). On a fixed n,Tn,T, the maps (A,Phi)|->S_(A,Phi)^(T_(CA)),M_(A,Phi)^(T_(CA)),L_(A,Phi)^(T_(CA))(A,\Phi)\mapsto S^{T_{\rm CA}}_{A,\Phi},\,M^{T_{\rm CA}}_{A,\Phi},\,L^{T_{\rm CA}}_{A,\Phi} are continuous in the product of discrete/trace topologies; hence any finite affine combination is continuous, and CL_(CA)=max_(T_(CA),u)F_(T_(CA),u)\mathrm{CL}_{\rm CA}=\max_{T_{\rm CA},u}F_{T_{\rm CA},u} is l.s.c.
Proof. Each is a cylinder-event probability or bounded expectation on a finite Markov chain parameterized by finitely many transition probabilities; continuity follows from finite-dimensionality. Max over a finite set preserves l.s.c. ∎
Lemma A1.1′ (concavity under mixing). If mu\mu is a distribution over base patterns {A_(j)}\{A_j\}, then A|->E_(mu)[F_(T_(CA),u)(A,Phi)]A\mapsto \mathbb E_\mu[F_{T_{\rm CA},u}(A,\Phi)] is affine in mu\mu; the pointwise maximum over (T_(CA),u)(T_{\rm CA},u) of affine maps is concave. Hence A|->CL_(CA)(A,Phi)A\mapsto \mathrm{CL}_{\rm CA}(A,\Phi) is concave in mu\mu.
Proof. Linearity in mu\mu is immediate (law of total expectation). Pointwise max of affine maps is concave. ∎
A1.3 Quantum toy: CL_(Q)\mathrm{CL}_{\rm Q} is l.s.c. and concave (under mixing)
Lemma A1.2 (l.s.c.). The map N|->||N(Delta)||_(1)\mathcal N\mapsto \|\mathcal N(\Delta)\|_1 is continuous in the diamond norm; taking i n f_(Phi in bar(P))\inf_{\Phi\in\overline{\mathscr P}} yields an l.s.c. functional CL_(Q)\mathrm{CL}_{\rm Q} on (A,Phi)(A,\Phi).
Proof. For fixed Delta\Delta, ||*(Delta)||_(1) <= ||*||_(diamond)||Delta||_(1)\|\cdot(\Delta)\|_1\le \|\cdot\|_\diamond\|\Delta\|_1. Continuity in ||*||_(diamond)\|\cdot\|_\diamond and the infimum over a compact poke class give l.s.c. ∎
Lemma A1.2′ (concavity under mixing). If A∼muA\sim \mu is a mixture of base channels {N_(A_(j),Phi)}\{\mathcal N_{A_j,\Phi}\}, then ||E_(mu)[N_(A,Phi)(Delta)]||_(1) <= E_(mu)[||N_(A,Phi)(Delta)||_(1)]\|\mathbb E_\mu[\mathcal N_{A,\Phi}(\Delta)]\|_1\le \mathbb E_\mu[\|\mathcal N_{A,\Phi}(\Delta)\|_1]. Thus A|->F_(Q)(A,Phi)A\mapsto F_{\rm Q}(A,\Phi) is concave in mu\mu, and so is A|->CL_(Q)(A,Phi)A\mapsto \mathrm{CL}_{\rm Q}(A,\Phi) after i n f _(Phi)\inf_\Phi.
Proof. Triangle inequality and Jensen for the norm; the affine dependence on mu\mu gives concavity. ∎
A1.4 Risk-sensitive worst-case limit
Proposition A1.2 (risk-sensitive limit). For either concrete CL and any poke law Pi\Pi, define CL_(beta)(A):=beta^(-1)log E_(Phi∼Pi)exp(betaCL(A,Phi))\mathrm{CL}_\beta(A):=\beta^{-1}\log\mathbb E_{\Phi\sim\Pi}\exp(\beta\,\mathrm{CL}(A,\Phi)). Then CL_(beta)↘i n f_(Phi in bar(P))CL(A,Phi)\mathrm{CL}_\beta\searrow \inf_{\Phi\in\overline{\mathscr P}}\mathrm{CL}(A,\Phi) pointwise and epi-converges as beta rarr oo\beta\to\infty.
Proof. Standard log-moment generating function convergence (Varadhan/Cramér type) on bounded functionals; epi-convergence follows from monotone convergence properties. ∎
A1.5 Robustness: no fine-tuning
Proposition A1.3 (equivalence class of CLs). Let F\mathcal F be the family
[
\mathcal F:=\Big{\mathrm{CL}s(A,\Phi):=\sup{T\in\mathscr T}\mathbb E\big[s_T(Z_{A,\Phi})\big]\ \Big|\ s_T:\mathcal Z_T\to\mathbb R\ bounded, concave, strictly increasing in success features\Big}.
]
On any bounded window and admissible poke cone, there exist constants 0 < c_(1) <= c_(2) < oo0<c_1\le c_2<\infty and increasing functions h_(1),h_(2)h_1,h_2 such that for any s,s^(')s,s' in the family,
uniformly on compact sets. Consequently, maximizing CL_(s)-sumlambda _(i)B_(i)\mathrm{CL}_s-\sum\lambda_i B_i or CL_(s^('))-sumlambda _(i)B_(i)\mathrm{CL}_{s'}-\sum\lambda_i B_i has the same maximizers in the Gamma\Gamma-limit, and the KKT multipliers (e.g. ℏ=lambda_(th)^(-1)\hbar=\lambda_{\rm th}^{-1}) agree.
Proof sketch. Bounded concave scores on finite observables are bi-Lipschitz equivalent up to increasing transforms on compact ranges; the sup over a finite T\mathscr T preserves equivalence. Stability of maximizers and multipliers follows from Gamma\Gamma-equivalence and envelope regularity established in Ch. 4–5. ∎
Corollary A1.4 (pointer robustness). For ss chosen as the quantum reliability score F_(Q)F_{\rm Q} or any strictly increasing concave transform thereof, leakage-budget minimizers align with the WW-eigenbasis (pointer basis), independent of the particular ss.